Exercise DiXED: Simplify: 1. \( \frac{x^{2}+x-6}{3 x^{2}-12 x} \div \frac{\frac{x^{3}-2 x^{2}}{x^{2}-16} \cdot \frac{1}{x+4}}{x^{2}+3 x+2}-\frac{1}{x^{2}+x-2}-\frac{1}{x^{2}-1} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(x^{3}-2x^{2}\right)}{\left(x^{2}-16\right)}\times \left(\frac{1}{\left(x+4\right)}\right)}{\left(x^{2}+3x+2\right)}-\left(\frac{1}{\left(x^{2}+x-2\right)}\right)-\left(\frac{1}{\left(x^{2}-1\right)}\right)\) - step1: Remove the parentheses: \(\frac{\frac{x^{3}-2x^{2}}{x^{2}-16}\times \left(\frac{1}{x+4}\right)}{x^{2}+3x+2}-\left(\frac{1}{x^{2}+x-2}\right)-\left(\frac{1}{x^{2}-1}\right)\) - step2: Remove the parentheses: \(\frac{\frac{x^{3}-2x^{2}}{x^{2}-16}\times \frac{1}{x+4}}{x^{2}+3x+2}-\left(\frac{1}{x^{2}+x-2}\right)-\left(\frac{1}{x^{2}-1}\right)\) - step3: Remove the parentheses: \(\frac{\frac{x^{3}-2x^{2}}{x^{2}-16}\times \frac{1}{x+4}}{x^{2}+3x+2}-\frac{1}{x^{2}+x-2}-\left(\frac{1}{x^{2}-1}\right)\) - step4: Remove the parentheses: \(\frac{\frac{x^{3}-2x^{2}}{x^{2}-16}\times \frac{1}{x+4}}{x^{2}+3x+2}-\frac{1}{x^{2}+x-2}-\frac{1}{x^{2}-1}\) - step5: Multiply the terms: \(\frac{\frac{x^{3}-2x^{2}}{\left(x^{2}-16\right)\left(x+4\right)}}{x^{2}+3x+2}-\frac{1}{x^{2}+x-2}-\frac{1}{x^{2}-1}\) - step6: Divide the terms: \(\frac{x^{3}-2x^{2}}{\left(x^{2}-16\right)\left(x+4\right)\left(x^{2}+3x+2\right)}-\frac{1}{x^{2}+x-2}-\frac{1}{x^{2}-1}\) - step7: Factor the expression: \(\frac{x^{3}-2x^{2}}{\left(x^{2}-16\right)\left(x+4\right)\left(x^{2}+3x+2\right)}-\frac{1}{\left(x-1\right)\left(x+2\right)}-\frac{1}{\left(x+1\right)\left(x-1\right)}\) - step8: Reduce fractions to a common denominator: \(\frac{\left(x^{3}-2x^{2}\right)\left(x-1\right)}{\left(x^{2}-16\right)\left(x+4\right)\left(x^{2}+3x+2\right)\left(x-1\right)}-\frac{\left(x^{2}-16\right)\left(x+4\right)\left(x+1\right)}{\left(x-1\right)\left(x+2\right)\left(x^{2}-16\right)\left(x+4\right)\left(x+1\right)}-\frac{\left(x+2\right)\left(x^{2}-16\right)\left(x+4\right)}{\left(x+1\right)\left(x-1\right)\left(x+2\right)\left(x^{2}-16\right)\left(x+4\right)}\) - step9: Rewrite the expression: \(\frac{\left(x^{3}-2x^{2}\right)\left(x-1\right)}{\left(x^{2}-16\right)\left(x+4\right)\left(x^{2}+3x+2\right)\left(x-1\right)}-\frac{\left(x^{2}-16\right)\left(x+4\right)\left(x+1\right)}{\left(x^{2}-16\right)\left(x+4\right)\left(x^{2}+3x+2\right)\left(x-1\right)}-\frac{\left(x+2\right)\left(x^{2}-16\right)\left(x+4\right)}{\left(x^{2}-16\right)\left(x+4\right)\left(x^{2}+3x+2\right)\left(x-1\right)}\) - step10: Transform the expression: \(\frac{\left(x^{3}-2x^{2}\right)\left(x-1\right)-\left(x^{2}-16\right)\left(x+4\right)\left(x+1\right)-\left(x+2\right)\left(x^{2}-16\right)\left(x+4\right)}{\left(x^{2}-16\right)\left(x+4\right)\left(x^{2}+3x+2\right)\left(x-1\right)}\) - step11: Multiply the terms: \(\frac{x^{4}-3x^{3}+2x^{2}-\left(x^{2}-16\right)\left(x+4\right)\left(x+1\right)-\left(x+2\right)\left(x^{2}-16\right)\left(x+4\right)}{\left(x^{2}-16\right)\left(x+4\right)\left(x^{2}+3x+2\right)\left(x-1\right)}\) - step12: Multiply the terms: \(\frac{x^{4}-3x^{3}+2x^{2}-\left(x^{4}+5x^{3}-12x^{2}-80x-64\right)-\left(x+2\right)\left(x^{2}-16\right)\left(x+4\right)}{\left(x^{2}-16\right)\left(x+4\right)\left(x^{2}+3x+2\right)\left(x-1\right)}\) - step13: Multiply the terms: \(\frac{x^{4}-3x^{3}+2x^{2}-\left(x^{4}+5x^{3}-12x^{2}-80x-64\right)-\left(x^{4}+6x^{3}-8x^{2}-96x-128\right)}{\left(x^{2}-16\right)\left(x+4\right)\left(x^{2}+3x+2\right)\left(x-1\right)}\) - step14: Calculate: \(\frac{-x^{4}-14x^{3}+22x^{2}+176x+192}{\left(x^{2}-16\right)\left(x+4\right)\left(x^{2}+3x+2\right)\left(x-1\right)}\) - step15: Simplify: \(\frac{-x^{4}-14x^{3}+22x^{2}+176x+192}{\left(x+4\right)^{2}\left(x-1\right)\left(x^{3}-x^{2}-10x-8\right)}\) - step16: Expand the expression: \(\frac{-x^{4}-14x^{3}+22x^{2}+176x+192}{x^{6}+6x^{5}-9x^{4}-102x^{3}-120x^{2}+96x+128}\) Calculate or simplify the expression \( (x^2+x-6)/(3*x^2-12*x) / ((x^3-2*x^2)/(x^2-16)*(1/(x+4))/(x^2+3*x+2)-(1/(x^2+x-2))-(1/(x^2-1))) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(x^{2}+x-6\right)}{\left(3x^{2}-12x\right)}}{\left(\frac{\frac{\left(x^{3}-2x^{2}\right)}{\left(x^{2}-16\right)}\times \left(\frac{1}{\left(x+4\right)}\right)}{\left(x^{2}+3x+2\right)}-\left(\frac{1}{\left(x^{2}+x-2\right)}\right)-\left(\frac{1}{\left(x^{2}-1\right)}\right)\right)}\) - step1: Remove the parentheses: \(\frac{\frac{x^{2}+x-6}{3x^{2}-12x}}{\frac{\frac{x^{3}-2x^{2}}{x^{2}-16}\times \left(\frac{1}{x+4}\right)}{x^{2}+3x+2}-\left(\frac{1}{x^{2}+x-2}\right)-\left(\frac{1}{x^{2}-1}\right)}\) - step2: Remove the parentheses: \(\frac{\frac{x^{2}+x-6}{3x^{2}-12x}}{\frac{\frac{x^{3}-2x^{2}}{x^{2}-16}\times \frac{1}{x+4}}{x^{2}+3x+2}-\left(\frac{1}{x^{2}+x-2}\right)-\left(\frac{1}{x^{2}-1}\right)}\) - step3: Remove the parentheses: \(\frac{\frac{x^{2}+x-6}{3x^{2}-12x}}{\frac{\frac{x^{3}-2x^{2}}{x^{2}-16}\times \frac{1}{x+4}}{x^{2}+3x+2}-\frac{1}{x^{2}+x-2}-\left(\frac{1}{x^{2}-1}\right)}\) - step4: Remove the parentheses: \(\frac{\frac{x^{2}+x-6}{3x^{2}-12x}}{\frac{\frac{x^{3}-2x^{2}}{x^{2}-16}\times \frac{1}{x+4}}{x^{2}+3x+2}-\frac{1}{x^{2}+x-2}-\frac{1}{x^{2}-1}}\) - step5: Multiply the terms: \(\frac{\frac{x^{2}+x-6}{3x^{2}-12x}}{\frac{\frac{x^{3}-2x^{2}}{\left(x^{2}-16\right)\left(x+4\right)}}{x^{2}+3x+2}-\frac{1}{x^{2}+x-2}-\frac{1}{x^{2}-1}}\) - step6: Divide the terms: \(\frac{\frac{x^{2}+x-6}{3x^{2}-12x}}{\frac{x^{3}-2x^{2}}{\left(x^{2}-16\right)\left(x+4\right)\left(x^{2}+3x+2\right)}-\frac{1}{x^{2}+x-2}-\frac{1}{x^{2}-1}}\) - step7: Subtract the terms: \(\frac{\frac{x^{2}+x-6}{3x^{2}-12x}}{\frac{-x^{4}-14x^{3}+22x^{2}+176x+192}{\left(x^{2}-16\right)\left(x+4\right)\left(x^{2}+3x+2\right)\left(x-1\right)}}\) - step8: Multiply by the reciprocal: \(\frac{x^{2}+x-6}{3x^{2}-12x}\times \frac{\left(x^{2}-16\right)\left(x+4\right)\left(x^{2}+3x+2\right)\left(x-1\right)}{-x^{4}-14x^{3}+22x^{2}+176x+192}\) - step9: Rewrite the expression: \(\frac{x^{2}+x-6}{3x\left(x-4\right)}\times \frac{\left(x-4\right)\left(x+4\right)\left(x+4\right)\left(x^{2}+3x+2\right)\left(x-1\right)}{-x^{4}-14x^{3}+22x^{2}+176x+192}\) - step10: Reduce the fraction: \(\frac{x^{2}+x-6}{3x}\times \frac{\left(x+4\right)\left(x+4\right)\left(x^{2}+3x+2\right)\left(x-1\right)}{-x^{4}-14x^{3}+22x^{2}+176x+192}\) - step11: Multiply the terms: \(\frac{\left(x^{2}+x-6\right)\left(x+4\right)^{2}\left(x^{2}+3x+2\right)\left(x-1\right)}{3x\left(-x^{4}-14x^{3}+22x^{2}+176x+192\right)}\) - step12: Expand the expression: \(\frac{x^{7}+11x^{6}+35x^{5}-7x^{4}-196x^{3}-196x^{2}+160x+192}{-3x^{5}-42x^{4}+66x^{3}+528x^{2}+576x}\) To simplify the expression \[ \frac{x^{2}+x-6}{3 x^{2}-12 x} \div \left( \frac{\frac{x^{3}-2 x^{2}}{x^{2}-16} \cdot \frac{1}{x+4}}{x^{2}+3 x+2}-\frac{1}{x^{2}+x-2}-\frac{1}{x^{2}-1} \right), \] we will break it down step by step. ### Step 1: Simplify the Denominator First, we simplify the denominator: \[ \frac{x^{3}-2 x^{2}}{x^{2}-16} \cdot \frac{1}{x+4} \div (x^{2}+3 x+2) - \frac{1}{x^{2}+x-2} - \frac{1}{x^{2}-1}. \] From our calculations, we found that: \[ \frac{x^{3}-2 x^{2}}{x^{2}-16} \cdot \frac{1}{x+4} = \frac{-x^{4}-14x^{3}+22x^{2}+176x+192}{x^{6}+6x^{5}-9x^{4}-102x^{3}-120x^{2}+96x+128}. \] ### Step 2: Substitute Back into the Original Expression Now, substituting this back into the original expression, we have: \[ \frac{x^{2}+x-6}{3 x^{2}-12 x} \div \left( \frac{-x^{4}-14x^{3}+22x^{2}+176x+192}{x^{6}+6x^{5}-9x^{4}-102x^{3}-120x^{2}+96x+128} \right) - \frac{1}{x^{2}+x-2} - \frac{1}{x^{2}-1}. \] ### Step 3: Simplify the Entire Expression Now, we can simplify the entire expression: \[ \frac{x^{2}+x-6}{3 x^{2}-12 x} \cdot \frac{x^{6}+6x^{5}-9x^{4}-102x^{3}-120x^{2}+96x+128}{-x^{4}-14x^{3}+22x^{2}+176x+192}. \] After performing the calculations, we find that the entire expression simplifies to: \[ \frac{x^{7}+11x^{6}+35x^{5}-7x^{4}-196x^{3}-196x^{2}+160x+192}{-3x^{5}-42x^{4}+66x^{3}+528x^{2}+576x}. \] ### Final Result Thus, the simplified form of the original expression is: \[ \frac{x^{7}+11x^{6}+35x^{5}-7x^{4}-196x^{3}-196x^{2}+160x+192}{-3x^{5}-42x^{4}+66x^{3}+528x^{2}+576x}. \]