(6) \( x^{2}+3 x+2 \) (7) \( x^{2}-10 x+24 \) (8) \( 3 x^{2}-15 x+12 \) (9) \( x^{3}+6 x^{2}+9 x \) (10) \( -x^{2}+9 x-20 \)

Factor the expression by following steps: - step0: Factor: \(x^{2}-10x+24\) - step1: Rewrite the expression: \(x^{2}+\left(-4-6\right)x+24\) - step2: Calculate: \(x^{2}-4x-6x+24\) - step3: Rewrite the expression: \(x\times x-x\times 4-6x+6\times 4\) - step4: Factor the expression: \(x\left(x-4\right)-6\left(x-4\right)\) - step5: Factor the expression: \(\left(x-6\right)\left(x-4\right)\) Factor the expression \( -x^{2}+9 x-20 \). Factor the expression by following steps: - step0: Factor: \(-x^{2}+9x-20\) - step1: Rewrite the expression: \(-x^{2}+\left(4+5\right)x-20\) - step2: Calculate: \(-x^{2}+4x+5x-20\) - step3: Rewrite the expression: \(-x\times x+x\times 4+5x-5\times 4\) - step4: Factor the expression: \(-x\left(x-4\right)+5\left(x-4\right)\) - step5: Factor the expression: \(\left(-x+5\right)\left(x-4\right)\) - step6: Factor the expression: \(-\left(x-5\right)\left(x-4\right)\) Factor the expression \( x^{2}+3 x+2 \). Factor the expression by following steps: - step0: Factor: \(x^{2}+3x+2\) - step1: Rewrite the expression: \(x^{2}+\left(2+1\right)x+2\) - step2: Calculate: \(x^{2}+2x+x+2\) - step3: Rewrite the expression: \(x\times x+x\times 2+x+2\) - step4: Factor the expression: \(x\left(x+2\right)+x+2\) - step5: Factor the expression: \(\left(x+1\right)\left(x+2\right)\) Factor the expression \( x^{3}+6 x^{2}+9 x \). Factor the expression by following steps: - step0: Factor: \(x^{3}+6x^{2}+9x\) - step1: Rewrite the expression: \(x\times x^{2}+x\times 6x+x\times 9\) - step2: Factor the expression: \(x\left(x^{2}+6x+9\right)\) - step3: Factor the expression: \(x\left(x+3\right)^{2}\) Factor the expression \( 3 x^{2}-15 x+12 \). Factor the expression by following steps: - step0: Factor: \(3x^{2}-15x+12\) - step1: Rewrite the expression: \(3x^{2}-3\times 5x+3\times 4\) - step2: Factor the expression: \(3\left(x^{2}-5x+4\right)\) - step3: Factor the expression: \(3\left(x-4\right)\left(x-1\right)\) Here are the factorizations for each of the given expressions: 1. **For \( x^{2}+3 x+2 \)**: \[ x^{2}+3 x+2 = (x+1)(x+2) \] 2. **For \( x^{2}-10 x+24 \)**: \[ x^{2}-10 x+24 = (x-6)(x-4) \] 3. **For \( 3 x^{2}-15 x+12 \)**: \[ 3 x^{2}-15 x+12 = 3(x-4)(x-1) \] 4. **For \( x^{3}+6 x^{2}+9 x \)**: \[ x^{3}+6 x^{2}+9 x = x(x+3)^{2} \] 5. **For \( -x^{2}+9 x-20 \)**: \[ -x^{2}+9 x-20 = -(x-5)(x-4) \] These factorizations simplify the expressions into products of polynomials.