\( \begin{array}{lll}\text { g) } \frac{-6}{x+2}-\frac{-x+3}{x}+\frac{x-1}{3 x} & \text { h) } & \frac{x^{2}+7 x}{x^{2}-4}+\frac{x-3}{2-x}-\frac{5}{x+2} \\ \text { i) } \frac{x}{x-1}+\frac{x+3}{x+1}-\frac{4}{1-x^{2}} & \text { j) } & \frac{a}{a x+a}-\frac{x-2}{x+1}+\frac{x+2}{x-1}\end{array} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{x}{\left(x-1\right)}+\frac{\left(x+3\right)}{\left(x+1\right)}-\frac{4}{\left(1-x^{2}\right)}\) - step1: Remove the parentheses: \(\frac{x}{x-1}+\frac{x+3}{x+1}-\frac{4}{1-x^{2}}\) - step2: Rewrite the fractions: \(\frac{x}{x-1}+\frac{x+3}{x+1}+\frac{4}{-1+x^{2}}\) - step3: Factor the expression: \(\frac{x}{x-1}+\frac{x+3}{x+1}+\frac{4}{\left(x+1\right)\left(x-1\right)}\) - step4: Reduce fractions to a common denominator: \(\frac{x\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}+\frac{\left(x+3\right)\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}+\frac{4}{\left(x+1\right)\left(x-1\right)}\) - step5: Rewrite the expression: \(\frac{x\left(x+1\right)}{\left(x+1\right)\left(x-1\right)}+\frac{\left(x+3\right)\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}+\frac{4}{\left(x+1\right)\left(x-1\right)}\) - step6: Transform the expression: \(\frac{x\left(x+1\right)+\left(x+3\right)\left(x-1\right)+4}{\left(x+1\right)\left(x-1\right)}\) - step7: Multiply the terms: \(\frac{x^{2}+x+\left(x+3\right)\left(x-1\right)+4}{\left(x+1\right)\left(x-1\right)}\) - step8: Multiply the terms: \(\frac{x^{2}+x+x^{2}+2x-3+4}{\left(x+1\right)\left(x-1\right)}\) - step9: Calculate: \(\frac{2x^{2}+3x+1}{\left(x+1\right)\left(x-1\right)}\) - step10: Factor the expression: \(\frac{\left(x+1\right)\left(2x+1\right)}{\left(x+1\right)\left(x-1\right)}\) - step11: Reduce the fraction: \(\frac{2x+1}{x-1}\) Calculate or simplify the expression \( a/(a*x + a) - (x - 2)/(x + 1) + (x + 2)/(x - 1) \). Simplify the expression by following steps: - step0: Solution: \(\frac{a}{\left(ax+a\right)}-\frac{\left(x-2\right)}{\left(x+1\right)}+\frac{\left(x+2\right)}{\left(x-1\right)}\) - step1: Remove the parentheses: \(\frac{a}{ax+a}-\frac{x-2}{x+1}+\frac{x+2}{x-1}\) - step2: Divide the terms: \(\frac{1}{x+1}-\frac{x-2}{x+1}+\frac{x+2}{x-1}\) - step3: Reduce fractions to a common denominator: \(\frac{x-1}{\left(x+1\right)\left(x-1\right)}-\frac{\left(x-2\right)\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}+\frac{\left(x+2\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}\) - step4: Rewrite the expression: \(\frac{x-1}{\left(x+1\right)\left(x-1\right)}-\frac{\left(x-2\right)\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}+\frac{\left(x+2\right)\left(x+1\right)}{\left(x+1\right)\left(x-1\right)}\) - step5: Transform the expression: \(\frac{x-1-\left(x-2\right)\left(x-1\right)+\left(x+2\right)\left(x+1\right)}{\left(x+1\right)\left(x-1\right)}\) - step6: Multiply the terms: \(\frac{x-1-\left(x^{2}-3x+2\right)+\left(x+2\right)\left(x+1\right)}{\left(x+1\right)\left(x-1\right)}\) - step7: Multiply the terms: \(\frac{x-1-\left(x^{2}-3x+2\right)+x^{2}+3x+2}{\left(x+1\right)\left(x-1\right)}\) - step8: Calculate: \(\frac{7x-1}{\left(x+1\right)\left(x-1\right)}\) - step9: Multiply the terms: \(\frac{7x-1}{x^{2}-1}\) Calculate or simplify the expression \( (x^2 + 7*x)/(x^2 - 4) + (x - 3)/(2 - x) - 5/(x + 2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x^{2}+7x\right)}{\left(x^{2}-4\right)}+\frac{\left(x-3\right)}{\left(2-x\right)}-\frac{5}{\left(x+2\right)}\) - step1: Remove the parentheses: \(\frac{x^{2}+7x}{x^{2}-4}+\frac{x-3}{2-x}-\frac{5}{x+2}\) - step2: Rewrite the fractions: \(\frac{x^{2}+7x}{x^{2}-4}-\frac{x-3}{-2+x}-\frac{5}{x+2}\) - step3: Factor the expression: \(\frac{x^{2}+7x}{\left(x+2\right)\left(x-2\right)}-\frac{x-3}{-2+x}-\frac{5}{x+2}\) - step4: Reduce fractions to a common denominator: \(\frac{x^{2}+7x}{\left(x+2\right)\left(x-2\right)}-\frac{\left(x-3\right)\left(x+2\right)}{\left(-2+x\right)\left(x+2\right)}-\frac{5\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}\) - step5: Rewrite the expression: \(\frac{x^{2}+7x}{\left(x+2\right)\left(x-2\right)}-\frac{\left(x-3\right)\left(x+2\right)}{\left(x+2\right)\left(x-2\right)}-\frac{5\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}\) - step6: Transform the expression: \(\frac{x^{2}+7x-\left(x-3\right)\left(x+2\right)-5\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}\) - step7: Multiply the terms: \(\frac{x^{2}+7x-\left(x^{2}-x-6\right)-5\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}\) - step8: Multiply the terms: \(\frac{x^{2}+7x-\left(x^{2}-x-6\right)-\left(5x-10\right)}{\left(x+2\right)\left(x-2\right)}\) - step9: Calculate: \(\frac{3x+16}{\left(x+2\right)\left(x-2\right)}\) - step10: Multiply the terms: \(\frac{3x+16}{x^{2}-4}\) Calculate or simplify the expression \( -6/(x+2) - (-x + 3)/x + (x - 1)/(3*x) \). Simplify the expression by following steps: - step0: Solution: \(\frac{-6}{\left(x+2\right)}-\frac{\left(-x+3\right)}{x}+\frac{\left(x-1\right)}{3x}\) - step1: Remove the parentheses: \(\frac{-6}{x+2}-\frac{-x+3}{x}+\frac{x-1}{3x}\) - step2: Use the rules for multiplication and division: \(\frac{-6}{x+2}+\frac{x-3}{x}+\frac{x-1}{3x}\) - step3: Rewrite the fraction: \(-\frac{6}{x+2}+\frac{x-3}{x}+\frac{x-1}{3x}\) - step4: Reduce fractions to a common denominator: \(-\frac{6\times 3x}{\left(x+2\right)\times 3x}+\frac{\left(x-3\right)\times 3\left(x+2\right)}{x\times 3\left(x+2\right)}+\frac{\left(x-1\right)\left(x+2\right)}{3x\left(x+2\right)}\) - step5: Reorder the terms: \(-\frac{6\times 3x}{3\left(x+2\right)x}+\frac{\left(x-3\right)\times 3\left(x+2\right)}{x\times 3\left(x+2\right)}+\frac{\left(x-1\right)\left(x+2\right)}{3x\left(x+2\right)}\) - step6: Reorder the terms: \(-\frac{6\times 3x}{3\left(x+2\right)x}+\frac{\left(x-3\right)\times 3\left(x+2\right)}{3x\left(x+2\right)}+\frac{\left(x-1\right)\left(x+2\right)}{3x\left(x+2\right)}\) - step7: Rewrite the expression: \(-\frac{6\times 3x}{3\left(x+2\right)x}+\frac{\left(x-3\right)\times 3\left(x+2\right)}{3\left(x+2\right)x}+\frac{\left(x-1\right)\left(x+2\right)}{3\left(x+2\right)x}\) - step8: Transform the expression: \(\frac{-6\times 3x+\left(x-3\right)\times 3\left(x+2\right)+\left(x-1\right)\left(x+2\right)}{3\left(x+2\right)x}\) - step9: Multiply the terms: \(\frac{-18x+\left(x-3\right)\times 3\left(x+2\right)+\left(x-1\right)\left(x+2\right)}{3\left(x+2\right)x}\) - step10: Multiply the terms: \(\frac{-18x+3x^{2}-3x-18+\left(x-1\right)\left(x+2\right)}{3\left(x+2\right)x}\) - step11: Multiply the terms: \(\frac{-18x+3x^{2}-3x-18+x^{2}+x-2}{3\left(x+2\right)x}\) - step12: Calculate: \(\frac{-20x+4x^{2}-20}{3\left(x+2\right)x}\) - step13: Calculate: \(\frac{-20x+4x^{2}-20}{3x^{2}+6x}\) Here are the simplified results for each expression: **g)** \[ \frac{-6}{x+2} - \frac{-x + 3}{x} + \frac{x - 1}{3x} = \frac{-20x + 4x^{2} - 20}{3x^{2} + 6x} \] **h)** \[ \frac{x^{2} + 7x}{x^{2} - 4} + \frac{x - 3}{2 - x} - \frac{5}{x + 2} = \frac{3x + 16}{x^{2} - 4} \] **i)** \[ \frac{x}{x - 1} + \frac{x + 3}{x + 1} - \frac{4}{1 - x^{2}} = \frac{2x + 1}{x - 1} \] **j)** \[ \frac{a}{ax + a} - \frac{x - 2}{x + 1} + \frac{x + 2}{x - 1} = \frac{7x - 1}{x^{2} - 1} \] If you need further assistance or explanations for any of these steps, feel free to ask!