(h) \( (x+1)^{2}=(x-1)(x+5) \) (j) \( 3-\frac{x-2}{5}=2 \) (l) \( \frac{x-2}{2}+\frac{x-1}{3}=2 \frac{5}{6} \) (n) \( \frac{x-1}{4}-\frac{x-2}{8}=\frac{x-4}{2}-1 \)

Solve the equation \( (x+1)^{2}=(x-1)(x+5) \). Solve the equation by following steps: - step0: Solve for \(x\): \(\left(x+1\right)^{2}=\left(x-1\right)\left(x+5\right)\) - step1: Multiply the terms: \(\left(x+1\right)^{2}=x^{2}+4x-5\) - step2: Expand the expression: \(x^{2}+2x+1=x^{2}+4x-5\) - step3: Cancel equal terms: \(2x+1=4x-5\) - step4: Move the expression to the left side: \(2x-4x=-5-1\) - step5: Add and subtract: \(-2x=-5-1\) - step6: Add and subtract: \(-2x=-6\) - step7: Change the signs: \(2x=6\) - step8: Divide both sides: \(\frac{2x}{2}=\frac{6}{2}\) - step9: Divide the numbers: \(x=3\) Solve the equation \( 3-\frac{x-2}{5}=2 \). Solve the equation by following steps: - step0: Solve for \(x\): \(3-\frac{x-2}{5}=2\) - step1: Multiply both sides of the equation by LCD: \(\left(3-\frac{x-2}{5}\right)\times 5=2\times 5\) - step2: Simplify the equation: \(17-x=10\) - step3: Move the constant to the right side: \(-x=10-17\) - step4: Subtract the numbers: \(-x=-7\) - step5: Change the signs: \(x=7\) Solve the equation \( \frac{x-2}{2}+\frac{x-1}{3}=2 \frac{5}{6} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{x-2}{2}+\frac{x-1}{3}=2\frac{5}{6}\) - step1: Convert the expressions: \(\frac{x-2}{2}+\frac{x-1}{3}=\frac{17}{6}\) - step2: Multiply both sides of the equation by LCD: \(\left(\frac{x-2}{2}+\frac{x-1}{3}\right)\times 6=\frac{17}{6}\times 6\) - step3: Simplify the equation: \(5x-8=17\) - step4: Move the constant to the right side: \(5x=17+8\) - step5: Add the numbers: \(5x=25\) - step6: Divide both sides: \(\frac{5x}{5}=\frac{25}{5}\) - step7: Divide the numbers: \(x=5\) Solve the equation \( \frac{x-1}{4}-\frac{x-2}{8}=\frac{x-4}{2}-1 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{x-1}{4}-\frac{x-2}{8}=\frac{x-4}{2}-1\) - step1: Multiply both sides of the equation by LCD: \(\left(\frac{x-1}{4}-\frac{x-2}{8}\right)\times 8=\left(\frac{x-4}{2}-1\right)\times 8\) - step2: Simplify the equation: \(x=4x-24\) - step3: Move the variable to the left side: \(x-4x=-24\) - step4: Subtract the terms: \(-3x=-24\) - step5: Change the signs: \(3x=24\) - step6: Divide both sides: \(\frac{3x}{3}=\frac{24}{3}\) - step7: Divide the numbers: \(x=8\) The solutions to the given equations are: 1. \( x = 3 \) 2. \( x = 7 \) 3. \( x = 5 \) 4. \( x = 8 \)