QUESTION 2 2.1) Simplify. 21.1) \( (2 x+3)\left(4 x^{2}-6 x+9\right) \) (3) 2.1.2) \( 2(x-y)^{2}+(x+2 y)(x-2 y) \) 2.2) \( \left(a^{2}-4\right)^{2}-\left(a^{2}+1\right)\left(a^{2}-1\right)+\left(2 a^{2}\right)^{2} \) 22) \( -3 x(3-x)\left(2 x^{2}-x-5\right) \) (2) \( \left(\frac{1}{2} x-3\right)\left(\frac{1}{4} x^{2}+\frac{3}{2} x+9\right) \)

Simplify the expression by following steps: - step0: Calculate: \(\left(\frac{1}{2}x-3\right)\left(\frac{1}{4}x^{2}+\frac{3}{2}x+9\right)\) - step1: Multiply the expression: \(2^{-3}\left(x^{3}-216\right)\) - step2: Rearrange the terms: \(2^{-3}x^{3}-27\) - step3: Evaluate the power: \(\frac{1}{8}x^{3}-27\) Expand the expression \( 2(x-y)^{2}+(x+2 y)(x-2 y) \) Simplify the expression by following steps: - step0: Add the terms: \(2\left(x-y\right)^{2}+\left(x+2y\right)\left(x-2y\right)\) - step1: Expand the expression: \(2x^{2}-4xy+2y^{2}+\left(x+2y\right)\left(x-2y\right)\) - step2: Expand the expression: \(2x^{2}-4xy+2y^{2}+x^{2}-4y^{2}\) - step3: Add the terms: \(3x^{2}-4xy-2y^{2}\) Expand the expression \( (2 x+3)(4 x^{2}-6 x+9) \) Simplify the expression by following steps: - step0: Simplify the product: \(\left(2x+3\right)\left(4x^{2}-6x+9\right)\) - step1: Simplify the product: \(8x^{3}+27\) Expand the expression \( (a^{2}-4)^{2}-(a^{2}+1)(a^{2}-1)+(2 a^{2})^{2} \) Simplify the expression by following steps: - step0: Simplify: \(\left(a^{2}-4\right)^{2}-\left(a^{2}+1\right)\left(a^{2}-1\right)+\left(2a^{2}\right)^{2}\) - step1: Multiply the terms: \(\left(a^{2}-4\right)^{2}+\left(-a^{2}-1\right)\left(a^{2}-1\right)+\left(2a^{2}\right)^{2}\) - step2: Expand the expression: \(a^{4}-8a^{2}+16+\left(-a^{2}-1\right)\left(a^{2}-1\right)+\left(2a^{2}\right)^{2}\) - step3: Expand the expression: \(a^{4}-8a^{2}+16+1-a^{4}+\left(2a^{2}\right)^{2}\) - step4: Rewrite the expression: \(a^{4}-8a^{2}+16+1-a^{4}+4a^{4}\) - step5: Calculate: \(4a^{4}-8a^{2}+17\) Expand the expression \( -3 x(3-x)(2 x^{2}-x-5) \) Simplify the expression by following steps: - step0: Calculate: \(-3x\left(3-x\right)\left(2x^{2}-x-5\right)\) - step1: Multiply the terms: \(\left(-9x+3x^{2}\right)\left(2x^{2}-x-5\right)\) - step2: Apply the distributive property: \(-9x\times 2x^{2}-\left(-9x\times x\right)-\left(-9x\times 5\right)+3x^{2}\times 2x^{2}-3x^{2}\times x-3x^{2}\times 5\) - step3: Multiply the terms: \(-18x^{3}-\left(-9x^{2}\right)-\left(-45x\right)+6x^{4}-3x^{3}-15x^{2}\) - step4: Remove the parentheses: \(-18x^{3}+9x^{2}+45x+6x^{4}-3x^{3}-15x^{2}\) - step5: Subtract the terms: \(-21x^{3}-6x^{2}+45x+6x^{4}\) Here are the simplified results for each expression: ### 2.1) Simplify #### 2.1.1) \[ (2x + 3)(4x^{2} - 6x + 9) = 8x^{3} + 27 \] #### 2.1.2) \[ 2(x - y)^{2} + (x + 2y)(x - 2y) = 3x^{2} - 4xy - 2y^{2} \] ### 2.2) \[ \left(a^{2} - 4\right)^{2} - \left(a^{2} + 1\right)\left(a^{2} - 1\right) + \left(2a^{2}\right)^{2} = 4a^{4} - 8a^{2} + 17 \] ### 2.3) \[ -3x(3 - x)(2x^{2} - x - 5) = -21x^{3} - 6x^{2} + 45x + 6x^{4} \] ### 2.4) \[ \left(\frac{1}{2}x - 3\right)\left(\frac{1}{4}x^{2} + \frac{3}{2}x + 9\right) = \frac{1}{8}x^{3} - 27 \] If you have any further questions or need additional assistance, feel free to ask!