\begin{tabular}{l} actorising and Simplifying \\ ool Maths Tuition \\ \( \frac{x^{3}+8}{x^{2}-4} \times \frac{5 x-10}{x^{2}-2 x+4} \) \\ \( \frac{x^{2}-5 x-6}{6 x-6} \times \frac{12 x-12}{x^{2}-1} \) \\ \( \frac{3}{a-4}+\frac{2}{a+3}-\frac{21}{a^{2}-a-12} \) \\ Solve for \( x \) \\ \( \frac{x}{x^{2}-3 x-4}=\frac{3}{x-4}-\frac{1}{2+2 x} \) \\ \hline\end{tabular}

To solve the given expressions, we will first simplify each expression and then solve for \( x \) in the last expression. 1. Simplify the first expression: \[ \frac{x^{3}+8}{x^{2}-4} \times \frac{5 x-10}{x^{2}-2 x+4} \] 2. Simplify the second expression: \[ \frac{x^{2}-5 x-6}{6 x-6} \times \frac{12 x-12}{x^{2}-1} \] 3. Simplify the third expression: \[ \frac{3}{a-4}+\frac{2}{a+3}-\frac{21}{a^{2}-a-12} \] 4. Solve for \( x \) in the last expression: \[ \frac{x}{x^{2}-3 x-4}=\frac{3}{x-4}-\frac{1}{2+2 x} \] Let's start by simplifying the expressions. Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(x^{3}+8\right)}{\left(x^{2}-4\right)}\times \left(5x-10\right)}{\left(x^{2}-2x+4\right)}\) - step1: Remove the parentheses: \(\frac{\frac{x^{3}+8}{x^{2}-4}\times \left(5x-10\right)}{x^{2}-2x+4}\) - step2: Divide the terms: \(\frac{\frac{x^{2}-2x+4}{x-2}\times \left(5x-10\right)}{x^{2}-2x+4}\) - step3: Multiply the terms: \(\frac{5\left(x^{2}-2x+4\right)}{x^{2}-2x+4}\) - step4: Reduce the fraction: \(5\) The first expression simplifies to \( 5 \). Now, let's simplify the second expression. Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(x^{2}-5x-6\right)}{\left(6x-6\right)}\times \left(12x-12\right)}{\left(x^{2}-1\right)}\) - step1: Remove the parentheses: \(\frac{\frac{x^{2}-5x-6}{6x-6}\times \left(12x-12\right)}{x^{2}-1}\) - step2: Multiply the terms: \(\frac{2\left(x^{2}-5x-6\right)}{x^{2}-1}\) - step3: Factor the expression: \(\frac{2\left(x+1\right)\left(x-6\right)}{\left(x+1\right)\left(x-1\right)}\) - step4: Reduce the fraction: \(\frac{2\left(x-6\right)}{x-1}\) - step5: Simplify: \(\frac{2x-12}{x-1}\) The second expression simplifies to \( \frac{2x-12}{x-1} \). Next, let's simplify the third expression. Simplify the expression by following steps: - step0: Solution: \(\frac{3}{\left(a-4\right)}+\frac{2}{\left(a+3\right)}-\frac{21}{\left(a^{2}-a-12\right)}\) - step1: Remove the parentheses: \(\frac{3}{a-4}+\frac{2}{a+3}-\frac{21}{a^{2}-a-12}\) - step2: Factor the expression: \(\frac{3}{a-4}+\frac{2}{a+3}-\frac{21}{\left(a-4\right)\left(a+3\right)}\) - step3: Reduce fractions to a common denominator: \(\frac{3\left(a+3\right)}{\left(a-4\right)\left(a+3\right)}+\frac{2\left(a-4\right)}{\left(a+3\right)\left(a-4\right)}-\frac{21}{\left(a-4\right)\left(a+3\right)}\) - step4: Rewrite the expression: \(\frac{3\left(a+3\right)}{\left(a-4\right)\left(a+3\right)}+\frac{2\left(a-4\right)}{\left(a-4\right)\left(a+3\right)}-\frac{21}{\left(a-4\right)\left(a+3\right)}\) - step5: Transform the expression: \(\frac{3\left(a+3\right)+2\left(a-4\right)-21}{\left(a-4\right)\left(a+3\right)}\) - step6: Multiply the terms: \(\frac{3a+9+2\left(a-4\right)-21}{\left(a-4\right)\left(a+3\right)}\) - step7: Multiply the terms: \(\frac{3a+9+2a-8-21}{\left(a-4\right)\left(a+3\right)}\) - step8: Calculate: \(\frac{5a-20}{\left(a-4\right)\left(a+3\right)}\) - step9: Factor the expression: \(\frac{5\left(a-4\right)}{\left(a-4\right)\left(a+3\right)}\) - step10: Reduce the fraction: \(\frac{5}{a+3}\) The third expression simplifies to \( \frac{5}{a+3} \). Finally, let's solve for \( x \) in the last expression. Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{x}{\left(x^{2}-3x-4\right)}=\frac{3}{\left(x-4\right)}-\frac{1}{\left(2+2x\right)}\) - step1: Find the domain: \(\frac{x}{\left(x^{2}-3x-4\right)}=\frac{3}{\left(x-4\right)}-\frac{1}{\left(2+2x\right)},x \in \left(-\infty,-1\right)\cup \left(-1,4\right)\cup \left(4,+\infty\right)\) - step2: Remove the parentheses: \(\frac{x}{x^{2}-3x-4}=\frac{3}{x-4}-\frac{1}{2+2x}\) - step3: Multiply both sides of the equation by LCD: \(\frac{x}{x^{2}-3x-4}\times 2\left(x^{2}-3x-4\right)=\left(\frac{3}{x-4}-\frac{1}{2+2x}\right)\times 2\left(x^{2}-3x-4\right)\) - step4: Simplify the equation: \(2x=10+5x\) - step5: Move the variable to the left side: \(2x-5x=10\) - step6: Subtract the terms: \(-3x=10\) - step7: Change the signs: \(3x=-10\) - step8: Divide both sides: \(\frac{3x}{3}=\frac{-10}{3}\) - step9: Divide the numbers: \(x=-\frac{10}{3}\) - step10: Check if the solution is in the defined range: \(x=-\frac{10}{3},x \in \left(-\infty,-1\right)\cup \left(-1,4\right)\cup \left(4,+\infty\right)\) - step11: Find the intersection: \(x=-\frac{10}{3}\) The solution for \( x \) in the last expression is \( x = -\frac{10}{3} \) or \( x = -3.\dot{3} \).