Solve x^2-3x-4/x^2-x-12*x^2-x-6/x^2+7x+10*frac{{x}^3+5{x}^2}{{x}^2-x}

$$\frac{ { x }^{ 2 } -3x-4 }{ { x }^{ 2 } -x-12 } \times \frac{ { x }^{ 2 } -x-6 }{ { x }^{ 2 } +7x+10 } \times \frac{ { x }^{ 3 } +5 { x }^{ 2 } }{ { x }^{ 2 } -x }$$

Evaluate

$$\frac{x\left(x-3\right)\left(x+1\right)}{\left(x-1\right)\left(x+3\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-3x-4}{x^{2}-x-12}.

$$\frac{\left(x-4\right)\left(x+1\right)}{\left(x-4\right)\left(x+3\right)}\times \frac{x^{2}-x-6}{x^{2}+7x+10}\times \frac{x^{3}+5x^{2}}{x^{2}-x}$$

Cancel out x-4 in both numerator and denominator.

$$\frac{x+1}{x+3}\times \frac{x^{2}-x-6}{x^{2}+7x+10}\times \frac{x^{3}+5x^{2}}{x^{2}-x}$$

Factor the expressions that are not already factored in \frac{x^{2}-x-6}{x^{2}+7x+10}.

$$\frac{x+1}{x+3}\times \frac{\left(x-3\right)\left(x+2\right)}{\left(x+2\right)\left(x+5\right)}\times \frac{x^{3}+5x^{2}}{x^{2}-x}$$

Cancel out x+2 in both numerator and denominator.

$$\frac{x+1}{x+3}\times \frac{x-3}{x+5}\times \frac{x^{3}+5x^{2}}{x^{2}-x}$$

Factor the expressions that are not already factored in \frac{x^{3}+5x^{2}}{x^{2}-x}.

$$\frac{x+1}{x+3}\times \frac{x-3}{x+5}\times \frac{\left(x+5\right)x^{2}}{x\left(x-1\right)}$$

Cancel out x in both numerator and denominator.

$$\frac{x+1}{x+3}\times \frac{x-3}{x+5}\times \frac{x\left(x+5\right)}{x-1}$$

Multiply \frac{x+1}{x+3} times \frac{x-3}{x+5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x+1\right)\left(x-3\right)}{\left(x+3\right)\left(x+5\right)}\times \frac{x\left(x+5\right)}{x-1}$$

Multiply \frac{\left(x+1\right)\left(x-3\right)}{\left(x+3\right)\left(x+5\right)} times \frac{x\left(x+5\right)}{x-1} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x+1\right)\left(x-3\right)x\left(x+5\right)}{\left(x+3\right)\left(x+5\right)\left(x-1\right)}$$

Cancel out x+5 in both numerator and denominator.

$$\frac{x\left(x-3\right)\left(x+1\right)}{\left(x-1\right)\left(x+3\right)}$$

Use the distributive property to multiply x by x-3.

$$\frac{\left(x^{2}-3x\right)\left(x+1\right)}{\left(x-1\right)\left(x+3\right)}$$

Use the distributive property to multiply x^{2}-3x by x+1 and combine like terms.

$$\frac{x^{3}-2x^{2}-3x}{\left(x-1\right)\left(x+3\right)}$$

Use the distributive property to multiply x-1 by x+3 and combine like terms.

$$\frac{x^{3}-2x^{2}-3x}{x^{2}+2x-3}$$

Expand

$$\frac{x^{3}-2x^{2}-3x}{\left(x-1\right)\left(x+3\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-3x-4}{x^{2}-x-12}.

$$\frac{\left(x-4\right)\left(x+1\right)}{\left(x-4\right)\left(x+3\right)}\times \frac{x^{2}-x-6}{x^{2}+7x+10}\times \frac{x^{3}+5x^{2}}{x^{2}-x}$$

Cancel out x-4 in both numerator and denominator.

$$\frac{x+1}{x+3}\times \frac{x^{2}-x-6}{x^{2}+7x+10}\times \frac{x^{3}+5x^{2}}{x^{2}-x}$$

Factor the expressions that are not already factored in \frac{x^{2}-x-6}{x^{2}+7x+10}.

$$\frac{x+1}{x+3}\times \frac{\left(x-3\right)\left(x+2\right)}{\left(x+2\right)\left(x+5\right)}\times \frac{x^{3}+5x^{2}}{x^{2}-x}$$

Cancel out x+2 in both numerator and denominator.

$$\frac{x+1}{x+3}\times \frac{x-3}{x+5}\times \frac{x^{3}+5x^{2}}{x^{2}-x}$$

Factor the expressions that are not already factored in \frac{x^{3}+5x^{2}}{x^{2}-x}.

$$\frac{x+1}{x+3}\times \frac{x-3}{x+5}\times \frac{\left(x+5\right)x^{2}}{x\left(x-1\right)}$$

Cancel out x in both numerator and denominator.

$$\frac{x+1}{x+3}\times \frac{x-3}{x+5}\times \frac{x\left(x+5\right)}{x-1}$$

Multiply \frac{x+1}{x+3} times \frac{x-3}{x+5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x+1\right)\left(x-3\right)}{\left(x+3\right)\left(x+5\right)}\times \frac{x\left(x+5\right)}{x-1}$$

Multiply \frac{\left(x+1\right)\left(x-3\right)}{\left(x+3\right)\left(x+5\right)} times \frac{x\left(x+5\right)}{x-1} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x+1\right)\left(x-3\right)x\left(x+5\right)}{\left(x+3\right)\left(x+5\right)\left(x-1\right)}$$

Cancel out x+5 in both numerator and denominator.

$$\frac{x\left(x-3\right)\left(x+1\right)}{\left(x-1\right)\left(x+3\right)}$$

Use the distributive property to multiply x by x-3.

$$\frac{\left(x^{2}-3x\right)\left(x+1\right)}{\left(x-1\right)\left(x+3\right)}$$

Use the distributive property to multiply x^{2}-3x by x+1 and combine like terms.

$$\frac{x^{3}-2x^{2}-3x}{\left(x-1\right)\left(x+3\right)}$$

Use the distributive property to multiply x-1 by x+3 and combine like terms.

$$\frac{x^{3}-2x^{2}-3x}{x^{2}+2x-3}$$