QUESTION 3 Eactorise the tollowing expressions complethly: \( \begin{array}{l}3.1 \quad(a-b)^{2}-3(b-a) \\ 3.2 \quad m^{3}-m^{2}-n m^{2}-n^{2} \\ \text { QUESTION } 4 \\ \text { Simplify the following expressions, assumung all denominatola are non-zeno: } \\ \frac{4 x^{\prime}-1}{3 x^{3}+10 x+3} \div \frac{6 x^{2}+5 x+1}{4 x^{2}+11 x-3} \times \frac{9 x^{2}+6 x+1}{8 x^{2}-6 x+1}\end{array} \)

Factor the expression by following steps: - step0: Factor: \(\left(a-b\right)^{2}-3\left(b-a\right)\) - step1: Factor the expression: \(\left(a-b+3\right)\left(a-b\right)\) Calculate or simplify the expression \( \frac{4 x^{\prime}-1}{3 x^{3}+10 x+3} \div \frac{6 x^{2}+5 x+1}{4 x^{2}+11 x-3} \times \frac{9 x^{2}+6 x+1}{8 x^{2}-6 x+1} \). Evaluate the derivative by following steps: - step0: Solution: \(\frac{4\times \left(x\right)^{\prime}-1}{3x^{3}+10x+3}\div \frac{6x^{2}+5x+1}{4x^{2}+11x-3}\times \frac{9x^{2}+6x+1}{8x^{2}-6x+1}\) - step1: Find the derivative: \(\frac{4\times 1-1}{3x^{3}+10x+3}\div \frac{6x^{2}+5x+1}{4x^{2}+11x-3}\times \frac{9x^{2}+6x+1}{8x^{2}-6x+1}\) - step2: Multiply: \(\frac{4-1}{3x^{3}+10x+3}\div \frac{6x^{2}+5x+1}{4x^{2}+11x-3}\times \frac{9x^{2}+6x+1}{8x^{2}-6x+1}\) - step3: Subtract the numbers: \(\frac{3}{3x^{3}+10x+3}\div \frac{6x^{2}+5x+1}{4x^{2}+11x-3}\times \frac{9x^{2}+6x+1}{8x^{2}-6x+1}\) - step4: Multiply by the reciprocal: \(\frac{3}{3x^{3}+10x+3}\times \frac{4x^{2}+11x-3}{6x^{2}+5x+1}\times \frac{9x^{2}+6x+1}{8x^{2}-6x+1}\) - step5: Multiply the terms: \(\frac{3\left(4x^{2}+11x-3\right)}{\left(3x^{3}+10x+3\right)\left(6x^{2}+5x+1\right)}\times \frac{9x^{2}+6x+1}{8x^{2}-6x+1}\) - step6: Rewrite the expression: \(\frac{3\left(4x-1\right)\left(x+3\right)}{\left(3x^{3}+10x+3\right)\left(6x^{2}+5x+1\right)}\times \frac{9x^{2}+6x+1}{\left(4x-1\right)\left(2x-1\right)}\) - step7: Reduce the fraction: \(\frac{3\left(x+3\right)}{\left(3x^{3}+10x+3\right)\left(6x^{2}+5x+1\right)}\times \frac{9x^{2}+6x+1}{2x-1}\) - step8: Rewrite the expression: \(\frac{3\left(x+3\right)}{\left(3x^{3}+10x+3\right)\left(3x+1\right)\left(2x+1\right)}\times \frac{\left(3x+1\right)^{2}}{2x-1}\) - step9: Reduce the fraction: \(\frac{3\left(x+3\right)}{\left(3x^{3}+10x+3\right)\left(2x+1\right)}\times \frac{3x+1}{2x-1}\) - step10: Multiply the terms: \(\frac{3\left(x+3\right)\left(3x+1\right)}{\left(3x^{3}+10x+3\right)\left(2x+1\right)\left(2x-1\right)}\) - step11: Calculate: \(\frac{9x^{2}+30x+9}{12x^{5}+37x^{3}+12x^{2}-10x-3}\) Let's go through the problems step by step. ### Question 3.1 We need to factor the expression \( (a-b)^{2}-3(b-a) \). 1. **Rewrite \( b-a \)**: Notice that \( b-a = -(a-b) \). Therefore, we can rewrite the expression as: \[ (a-b)^{2} - 3(-1)(a-b) = (a-b)^{2} + 3(a-b) \] 2. **Factor out \( (a-b) \)**: \[ (a-b)((a-b) + 3) = (a-b)(a-b+3) \] Thus, the completely factored form is: \[ (a-b+3)(a-b) \] ### Question 3.2 We need to factor the expression \( m^{3}-m^{2}-n m^{2}-n^{2} \). 1. **Group the terms**: \[ m^{3} - m^{2} - n m^{2} - n^{2} = m^{2}(m - 1 - n) - n^{2} \] 2. **Notice that this is a difference of squares**: \[ = m^{2}(m - 1 - n) - n^{2} = (m^{2} - n^{2})(m - 1 - n) \] 3. **Factor the difference of squares**: \[ = (m - n)(m + n)(m - 1 - n) \] However, I encountered an error while trying to factor this expression completely. Let's focus on the first part, which is \( m^{2} - n^{2} \). ### Question 4 We need to simplify the expression: \[ \frac{4 x^{\prime}-1}{3 x^{3}+10 x+3} \div \frac{6 x^{2}+5 x+1}{4 x^{2}+11 x-3} \times \frac{9 x^{2}+6 x+1}{8 x^{2}-6 x+1} \] The simplified result is: \[ \frac{9x^{2}+30x+9}{12x^{5}+37x^{3}+12x^{2}-10x-3} \] ### Summary of Results - **3.1**: The factored form is \( (a-b+3)(a-b) \). - **3.2**: The expression could not be factored completely, but we have \( (m - n)(m + n)(m - 1 - n) \) as part of the factorization. - **4**: The simplified expression is \( \frac{9x^{2}+30x+9}{12x^{5}+37x^{3}+12x^{2}-10x-3} \).