5. Factorice completamente las siguientes expresiones. 1 a) \( x^{4}+x-x^{3} y-y \) b) \( x^{3}-x-x^{2} y+y \) c) \( 6 x^{2}+x y-y^{2} \) d) \( a^{2}-b^{3}+2 b^{3} x^{2}-2 a^{2} x^{2} \) e) \( a^{2}+9 a+20 \) f) \( a^{2}-7 a+12 \) g) \( a^{2}-6 a+9 \) h) \( 6 x^{2}-x-2 \) i) \( 6 x^{2}+7 x y-3 y^{2} \) j) \( m^{4}+m^{2} n^{2}+n^{4} \) k) \( 15+14 x-8 x^{2} \) l) \( x^{6}+x^{3}-2 \) m) \( 2 \sqrt[3]{x^{2}}+5 \sqrt[3]{x}+2 \) n) \( 4 a^{2 n}-b^{2} \) n) \( x^{8}-y^{8} \) o) \( (\sqrt{6}-\sqrt{2}) \)

Factor the expression by following steps: - step0: Factor: \(a^{2}-6a+9\) - step1: Factor the expression: \(\left(a-3\right)^{2}\) Factor the expression \( 6 x^{2}+x y-y^{2 \). Factor the expression by following steps: - step0: Factor: \(6x^{2}+xy-y^{2}\) - step1: Rewrite the expression: \(6x^{2}+\left(3-2\right)xy-y^{2}\) - step2: Calculate: \(6x^{2}+3xy-2yx-y^{2}\) - step3: Rewrite the expression: \(3x\times 2x+3xy-y\times 2x-y\times y\) - step4: Factor the expression: \(3x\left(2x+y\right)-y\left(2x+y\right)\) - step5: Factor the expression: \(\left(3x-y\right)\left(2x+y\right)\) Factor the expression \( a^{2}+9 a+20 \). Factor the expression by following steps: - step0: Factor: \(a^{2}+9a+20\) - step1: Rewrite the expression: \(a^{2}+\left(5+4\right)a+20\) - step2: Calculate: \(a^{2}+5a+4a+20\) - step3: Rewrite the expression: \(a\times a+a\times 5+4a+4\times 5\) - step4: Factor the expression: \(a\left(a+5\right)+4\left(a+5\right)\) - step5: Factor the expression: \(\left(a+4\right)\left(a+5\right)\) Factor the expression \( a^{2}-b^{3}+2 b^{3} x^{2}-2 a^{2} x^{2} \). Factor the expression by following steps: - step0: Factor: \(a^{2}-b^{3}+2b^{3}x^{2}-2a^{2}x^{2}\) - step1: Calculate: \(2b^{3}x^{2}-b^{3}-2a^{2}x^{2}+a^{2}\) - step2: Rewrite the expression: \(b^{3}\times 2x^{2}-b^{3}-a^{2}\times 2x^{2}+a^{2}\) - step3: Factor the expression: \(b^{3}\left(2x^{2}-1\right)-a^{2}\left(2x^{2}-1\right)\) - step4: Factor the expression: \(\left(b^{3}-a^{2}\right)\left(2x^{2}-1\right)\) Factor the expression \( a^{2}-7 a+12 \). Factor the expression by following steps: - step0: Factor: \(a^{2}-7a+12\) - step1: Rewrite the expression: \(a^{2}+\left(-3-4\right)a+12\) - step2: Calculate: \(a^{2}-3a-4a+12\) - step3: Rewrite the expression: \(a\times a-a\times 3-4a+4\times 3\) - step4: Factor the expression: \(a\left(a-3\right)-4\left(a-3\right)\) - step5: Factor the expression: \(\left(a-4\right)\left(a-3\right)\) Factor the expression \( 15+14 x-8 x^{2 \). Factor the expression by following steps: - step0: Factor: \(15+14x-8x^{2}\) - step1: Rewrite the expression: \(15+\left(20-6\right)x-8x^{2}\) - step2: Calculate: \(15+20x-6x-8x^{2}\) - step3: Rewrite the expression: \(5\times 3+5\times 4x-2x\times 3-2x\times 4x\) - step4: Factor the expression: \(5\left(3+4x\right)-2x\left(3+4x\right)\) - step5: Factor the expression: \(\left(5-2x\right)\left(3+4x\right)\) Factor the expression \( x^{4}+x-x^{3} y-y \). Factor the expression by following steps: - step0: Factor: \(x^{4}+x-x^{3}y-y\) - step1: Evaluate: \(x^{4}-x^{3}y+x-y\) - step2: Calculate: \(x^{4}+x^{3}-x^{3}y-x^{2}y-x^{3}-x^{2}+x^{2}y+xy+x^{2}+x-yx-y\) - step3: Rewrite the expression: \(x^{2}\times x^{2}+x^{2}\times x-x^{2}yx-x^{2}y-x\times x^{2}-x\times x+xyx+xy+x^{2}+x-yx-y\) - step4: Factor the expression: \(x^{2}\left(x^{2}+x-yx-y\right)-x\left(x^{2}+x-yx-y\right)+x^{2}+x-yx-y\) - step5: Factor the expression: \(\left(x^{2}-x+1\right)\left(x^{2}+x-yx-y\right)\) - step6: Factor the expression: \(\left(x^{2}-x+1\right)\left(x-y\right)\left(x+1\right)\) Factor the expression \( 6 x^{2}-x-2 \). Factor the expression by following steps: - step0: Factor: \(6x^{2}-x-2\) - step1: Rewrite the expression: \(6x^{2}+\left(-4+3\right)x-2\) - step2: Calculate: \(6x^{2}-4x+3x-2\) - step3: Rewrite the expression: \(2x\times 3x-2x\times 2+3x-2\) - step4: Factor the expression: \(2x\left(3x-2\right)+3x-2\) - step5: Factor the expression: \(\left(2x+1\right)\left(3x-2\right)\) Factor the expression \( 6 x^{2}+7 x y-3 y^{2 \). Factor the expression by following steps: - step0: Factor: \(6x^{2}+7xy-3y^{2}\) - step1: Rewrite the expression: \(6x^{2}+\left(9-2\right)xy-3y^{2}\) - step2: Calculate: \(6x^{2}+9xy-2yx-3y^{2}\) - step3: Rewrite the expression: \(3x\times 2x+3x\times 3y-y\times 2x-y\times 3y\) - step4: Factor the expression: \(3x\left(2x+3y\right)-y\left(2x+3y\right)\) - step5: Factor the expression: \(\left(3x-y\right)\left(2x+3y\right)\) Factor the expression \( m^{4}+m^{2} n^{2}+n^{4 \). Factor the expression by following steps: - step0: Factor: \(m^{4}+m^{2}n^{2}+n^{4}\) - step1: Calculate: \(m^{4}-m^{3}n+m^{2}n^{2}+m^{3}n-m^{2}n^{2}+mn^{3}+n^{2}m^{2}-n^{3}m+n^{4}\) - step2: Rewrite the expression: \(m^{2}\times m^{2}-m^{2}\times mn+m^{2}n^{2}+mnm^{2}-mnmn+mn\times n^{2}+n^{2}m^{2}-n^{2}mn+n^{2}\times n^{2}\) - step3: Factor the expression: \(m^{2}\left(m^{2}-mn+n^{2}\right)+mn\left(m^{2}-mn+n^{2}\right)+n^{2}\left(m^{2}-mn+n^{2}\right)\) - step4: Factor the expression: \(\left(m^{2}+mn+n^{2}\right)\left(m^{2}-mn+n^{2}\right)\) Factor the expression \( 2 \sqrt[3]{x^{2}}+5 \sqrt[3]{x}+2 \). Factor the expression by following steps: - step0: Factor: \(2\sqrt[3]{x^{2}}+5\sqrt[3]{x}+2\) - step1: Evaluate: \(2x^{\frac{2}{3}}+5x^{\frac{1}{3}}+2\) - step2: Rewrite the expression: \(2x^{\frac{2}{3}}+\left(1+4\right)x^{\frac{1}{3}}+2\) - step3: Calculate: \(2x^{\frac{2}{3}}+x^{\frac{1}{3}}+4x^{\frac{1}{3}}+2\) - step4: Rewrite the expression: \(x^{\frac{1}{3}}\times 2x^{\frac{1}{3}}+x^{\frac{1}{3}}+2\times 2x^{\frac{1}{3}}+2\) - step5: Factor the expression: \(x^{\frac{1}{3}}\left(2x^{\frac{1}{3}}+1\right)+2\left(2x^{\frac{1}{3}}+1\right)\) - step6: Factor the expression: \(\left(x^{\frac{1}{3}}+2\right)\left(2x^{\frac{1}{3}}+1\right)\) - step7: Rewrite the expression: \(\left(\sqrt[3]{x}+2\right)\left(2\sqrt[3]{x}+1\right)\) Factor the expression \( x^{6}+x^{3}-2 \). Factor the expression by following steps: - step0: Factor: \(x^{6}+x^{3}-2\) - step1: Calculate: \(x^{6}+2x^{3}+x^{5}+2x^{2}+x^{4}+2x-x^{5}-2x^{2}-x^{4}-2x-x^{3}-2\) - step2: Rewrite the expression: \(x\times x^{5}+x\times 2x^{2}+x\times x^{4}+x\times 2x+x\times x^{3}+x\times 2-x^{5}-2x^{2}-x^{4}-2x-x^{3}-2\) - step3: Factor the expression: \(x\left(x^{5}+2x^{2}+x^{4}+2x+x^{3}+2\right)-\left(x^{5}+2x^{2}+x^{4}+2x+x^{3}+2\right)\) - step4: Factor the expression: \(\left(x-1\right)\left(x^{5}+2x^{2}+x^{4}+2x+x^{3}+2\right)\) - step5: Factor the expression: \(\left(x-1\right)\left(x^{2}+x+1\right)\left(x^{3}+2\right)\) Factor the expression \( x^{3}-x-x^{2} y+y \). Factor the expression by following steps: - step0: Factor: \(x^{3}-x-x^{2}y+y\) - step1: Calculate: \(x^{3}-x-yx^{2}+y\) - step2: Rewrite the expression: \(x\times x^{2}-x-yx^{2}+y\) - step3: Factor the expression: \(x\left(x^{2}-1\right)-y\left(x^{2}-1\right)\) - step4: Factor the expression: \(\left(x-y\right)\left(x^{2}-1\right)\) - step5: Factor the expression: \(\left(x-y\right)\left(x+1\right)\left(x-1\right)\) Factor the expression \( 4 a^{2 n}-b^{2 \). Factor the expression by following steps: - step0: Factor: \(4a^{2n}-b^{2}\) - step1: Evaluate: \(4\left(a^{n}\right)^{2}-b^{2}\) - step2: Factor the expression: \(\left(2a^{n}+b\right)\left(2a^{n}-b\right)\) Factor the expression \( x^{8}-y^{8 \). Factor the expression by following steps: - step0: Factor: \(x^{8}-y^{8}\) - step1: Rewrite the expression: \(\left(x^{4}\right)^{2}-\left(y^{4}\right)^{2}\) - step2: Factor the expression: \(\left(x^{4}-y^{4}\right)\left(x^{4}+y^{4}\right)\) - step3: Evaluate: \(\left(x-y\right)\left(x+y\right)\left(x^{2}+y^{2}\right)\left(x^{4}+y^{4}\right)\) Aquí están las factorizaciones completas de las expresiones dadas: a) \( x^{4}+x-x^{3} y-y \) se factoriza como: \[ (x^{2}-x+1)(x-y)(x+1) \] b) \( x^{3}-x-x^{2} y+y \) se factoriza como: \[ (x-y)(x+1)(x-1) \] c) \( 6 x^{2}+x y-y^{2} \) se factoriza como: \[ (3x-y)(2x+y) \] d) \( a^{2}-b^{3}+2 b^{3} x^{2}-2 a^{2} x^{2} \) se factoriza como: \[ (b^{3}-a^{2})(2x^{2}-1) \] e) \( a^{2}+9 a+20 \) se factoriza como: \[ (a+4)(a+5) \] f) \( a^{2}-7 a+12 \) se factoriza como: \[ (a-4)(a-3) \] g) \( a^{2}-6 a+9 \) se factoriza como: \[ (a-3)^{2} \] h) \( 6 x^{2}-x-2 \) se factoriza como: \[ (2x+1)(3x-2) \] i) \( 6 x^{2}+7 x y-3 y^{2} \) se factoriza como: \[ (3x-y)(2x+3y) \] j) \( m^{4}+m^{2} n^{2}+n^{4} \) se factoriza como: \[ (m^{2}+mn+n^{2})(m^{2}-mn+n^{2}) \] k) \( 15+14 x-8 x^{2} \) se factoriza como: \[ (5-2x)(3+4x) \] l) \( x^{6}+x^{3}-2 \) se factoriza como: \[ (x-1)(x^{2}+x+1)(x^{3}+2) \] m) \( 2 \sqrt[3]{x^{2}}+5 \sqrt[3]{x}+2 \) se factoriza como: \[ (\sqrt[3]{x}+2)(2\sqrt[3]{x}+1) \] n) \( 4 a^{2 n}-b^{2} \) se factoriza como: \[ (2a^{n}+b)(2a^{n}-b) \] o) \( x^{8}-y^{8} \) se factoriza como: \[ (x-y)(x+y)(x^{2}+y^{2})(x^{4}+y^{4}) \] p) \( (\sqrt{6}-\sqrt{2}) \) no se puede factorizar más allá de su forma actual. Si necesitas más ayuda o aclaraciones sobre alguna de las factorizaciones, no dudes en preguntar.