Solve frac{x^{3}-5x^{2}}{x^{3}-25x}divfrac{x^{2}+3x}{x^{2}+5x+6}+x^2+3x-4/x^2+6x+8*x^2-x-6/x^2-6x+5

$$\frac { x ^ { 3 } - 5 x ^ { 2 } } { x ^ { 3 } - 25 x } \div \frac { x ^ { 2 } + 3 x } { x ^ { 2 } + 5 x + 6 } + \frac { x ^ { 2 } + 3 x - 4 } { x ^ { 2 } + 6 x + 8 } \cdot \frac { x ^ { 2 } - x - 6 } { x ^ { 2 } - 6 x + 5 }$$

Evaluate

$$\frac{2x^{2}-x-25}{x^{2}-25}$$

Short Solution Steps

Divide \frac{x^{3}-5x^{2}}{x^{3}-25x} by \frac{x^{2}+3x}{x^{2}+5x+6} by multiplying \frac{x^{3}-5x^{2}}{x^{3}-25x} by the reciprocal of \frac{x^{2}+3x}{x^{2}+5x+6}.

$$\frac{\left(x^{3}-5x^{2}\right)\left(x^{2}+5x+6\right)}{\left(x^{3}-25x\right)\left(x^{2}+3x\right)}+\frac{x^{2}+3x-4}{x^{2}+6x+8}\times \frac{x^{2}-x-6}{x^{2}-6x+5}$$

Factor the expressions that are not already factored in \frac{\left(x^{3}-5x^{2}\right)\left(x^{2}+5x+6\right)}{\left(x^{3}-25x\right)\left(x^{2}+3x\right)}.

$$\frac{\left(x-5\right)\left(x+2\right)\left(x+3\right)x^{2}}{\left(x-5\right)\left(x+3\right)\left(x+5\right)x^{2}}+\frac{x^{2}+3x-4}{x^{2}+6x+8}\times \frac{x^{2}-x-6}{x^{2}-6x+5}$$

Cancel out \left(x-5\right)\left(x+3\right)x^{2} in both numerator and denominator.

$$\frac{x+2}{x+5}+\frac{x^{2}+3x-4}{x^{2}+6x+8}\times \frac{x^{2}-x-6}{x^{2}-6x+5}$$

Factor the expressions that are not already factored in \frac{x^{2}+3x-4}{x^{2}+6x+8}.

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

Cancel out x+4 in both numerator and denominator.

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

Multiply \frac{x-1}{x+2} times \frac{x^{2}-x-6}{x^{2}-6x+5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x+2}{x+5}+\frac{\left(x-1\right)\left(x^{2}-x-6\right)}{\left(x+2\right)\left(x^{2}-6x+5\right)}$$

Factor \left(x+2\right)\left(x^{2}-6x+5\right).

$$\frac{x+2}{x+5}+\frac{\left(x-1\right)\left(x^{2}-x-6\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+5 and \left(x-5\right)\left(x-1\right)\left(x+2\right) is \left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right). Multiply \frac{x+2}{x+5} times \frac{\left(x-5\right)\left(x-1\right)\left(x+2\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}. Multiply \frac{\left(x-1\right)\left(x^{2}-x-6\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)} times \frac{x+5}{x+5}.

$$\frac{\left(x+2\right)\left(x-5\right)\left(x-1\right)\left(x+2\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)}+\frac{\left(x-1\right)\left(x^{2}-x-6\right)\left(x+5\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)}$$

Since \frac{\left(x+2\right)\left(x-5\right)\left(x-1\right)\left(x+2\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)} and \frac{\left(x-1\right)\left(x^{2}-x-6\right)\left(x+5\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+2\right)\left(x-5\right)\left(x-1\right)\left(x+2\right)+\left(x-1\right)\left(x^{2}-x-6\right)\left(x+5\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)}$$

Do the multiplications in \left(x+2\right)\left(x-5\right)\left(x-1\right)\left(x+2\right)+\left(x-1\right)\left(x^{2}-x-6\right)\left(x+5\right).

$$\frac{x^{4}-4x^{3}-7x^{2}+10x+2x^{3}-8x^{2}-14x+20+x^{4}+4x^{3}-11x^{2}-30x-x^{3}-4x^{2}+11x+30}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)}$$

Combine like terms in x^{4}-4x^{3}-7x^{2}+10x+2x^{3}-8x^{2}-14x+20+x^{4}+4x^{3}-11x^{2}-30x-x^{3}-4x^{2}+11x+30.

$$\frac{2x^{4}+x^{3}-30x^{2}-23x+50}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)}$$

Factor the expressions that are not already factored in \frac{2x^{4}+x^{3}-30x^{2}-23x+50}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)}.

$$\frac{2\left(x-1\right)\left(x+2\right)\left(x^{2}-\frac{1}{2}x-\frac{25}{2}\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)}$$

Cancel out \left(x-1\right)\left(x+2\right) in both numerator and denominator.

$$\frac{2\left(x^{2}-\frac{1}{2}x-\frac{25}{2}\right)}{\left(x-5\right)\left(x+5\right)}$$

Expand \left(x-5\right)\left(x+5\right).

$$\frac{2\left(x^{2}-\frac{1}{2}x-\frac{25}{2}\right)}{x^{2}-25}$$

Use the distributive property to multiply 2 by x^{2}-\frac{1}{2}x-\frac{25}{2}.

$$\frac{2x^{2}-x-25}{x^{2}-25}$$

Expand

$$\frac{2x^{2}-x-25}{x^{2}-25}$$

Short Solution Steps

Divide \frac{x^{3}-5x^{2}}{x^{3}-25x} by \frac{x^{2}+3x}{x^{2}+5x+6} by multiplying \frac{x^{3}-5x^{2}}{x^{3}-25x} by the reciprocal of \frac{x^{2}+3x}{x^{2}+5x+6}.

$$\frac{\left(x^{3}-5x^{2}\right)\left(x^{2}+5x+6\right)}{\left(x^{3}-25x\right)\left(x^{2}+3x\right)}+\frac{x^{2}+3x-4}{x^{2}+6x+8}\times \frac{x^{2}-x-6}{x^{2}-6x+5}$$

Factor the expressions that are not already factored in \frac{\left(x^{3}-5x^{2}\right)\left(x^{2}+5x+6\right)}{\left(x^{3}-25x\right)\left(x^{2}+3x\right)}.

$$\frac{\left(x-5\right)\left(x+2\right)\left(x+3\right)x^{2}}{\left(x-5\right)\left(x+3\right)\left(x+5\right)x^{2}}+\frac{x^{2}+3x-4}{x^{2}+6x+8}\times \frac{x^{2}-x-6}{x^{2}-6x+5}$$

Cancel out \left(x-5\right)\left(x+3\right)x^{2} in both numerator and denominator.

$$\frac{x+2}{x+5}+\frac{x^{2}+3x-4}{x^{2}+6x+8}\times \frac{x^{2}-x-6}{x^{2}-6x+5}$$

Factor the expressions that are not already factored in \frac{x^{2}+3x-4}{x^{2}+6x+8}.

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

Cancel out x+4 in both numerator and denominator.

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

Multiply \frac{x-1}{x+2} times \frac{x^{2}-x-6}{x^{2}-6x+5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x+2}{x+5}+\frac{\left(x-1\right)\left(x^{2}-x-6\right)}{\left(x+2\right)\left(x^{2}-6x+5\right)}$$

Factor \left(x+2\right)\left(x^{2}-6x+5\right).

$$\frac{x+2}{x+5}+\frac{\left(x-1\right)\left(x^{2}-x-6\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+5 and \left(x-5\right)\left(x-1\right)\left(x+2\right) is \left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right). Multiply \frac{x+2}{x+5} times \frac{\left(x-5\right)\left(x-1\right)\left(x+2\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}. Multiply \frac{\left(x-1\right)\left(x^{2}-x-6\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)} times \frac{x+5}{x+5}.

$$\frac{\left(x+2\right)\left(x-5\right)\left(x-1\right)\left(x+2\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)}+\frac{\left(x-1\right)\left(x^{2}-x-6\right)\left(x+5\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)}$$

Since \frac{\left(x+2\right)\left(x-5\right)\left(x-1\right)\left(x+2\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)} and \frac{\left(x-1\right)\left(x^{2}-x-6\right)\left(x+5\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+2\right)\left(x-5\right)\left(x-1\right)\left(x+2\right)+\left(x-1\right)\left(x^{2}-x-6\right)\left(x+5\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)}$$

Do the multiplications in \left(x+2\right)\left(x-5\right)\left(x-1\right)\left(x+2\right)+\left(x-1\right)\left(x^{2}-x-6\right)\left(x+5\right).

$$\frac{x^{4}-4x^{3}-7x^{2}+10x+2x^{3}-8x^{2}-14x+20+x^{4}+4x^{3}-11x^{2}-30x-x^{3}-4x^{2}+11x+30}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)}$$

Combine like terms in x^{4}-4x^{3}-7x^{2}+10x+2x^{3}-8x^{2}-14x+20+x^{4}+4x^{3}-11x^{2}-30x-x^{3}-4x^{2}+11x+30.

$$\frac{2x^{4}+x^{3}-30x^{2}-23x+50}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)}$$

Factor the expressions that are not already factored in \frac{2x^{4}+x^{3}-30x^{2}-23x+50}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)}.

$$\frac{2\left(x-1\right)\left(x+2\right)\left(x^{2}-\frac{1}{2}x-\frac{25}{2}\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+5\right)}$$

Cancel out \left(x-1\right)\left(x+2\right) in both numerator and denominator.

$$\frac{2\left(x^{2}-\frac{1}{2}x-\frac{25}{2}\right)}{\left(x-5\right)\left(x+5\right)}$$

Expand \left(x-5\right)\left(x+5\right).

$$\frac{2\left(x^{2}-\frac{1}{2}x-\frac{25}{2}\right)}{x^{2}-25}$$

Use the distributive property to multiply 2 by x^{2}-\frac{1}{2}x-\frac{25}{2}.

$$\frac{2x^{2}-x-25}{x^{2}-25}$$