Solve (x^3-125/x^2+5x+25)div(x/x-2+2/x+3-3/x^2+x-6)

$$( \frac { x ^ { 3 } - 125 } { x ^ { 2 } + 5 x + 25 } ) \div ( \frac { x } { x - 2 } + \frac { 2 } { x + 3 } - \frac { 3 } { x ^ { 2 } + x - 6 } )$$

Evaluate

$$\frac{\left(x-5\right)\left(x-2\right)\left(x+3\right)}{x^{2}+5x-7}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{3}-125}{x^{2}+5x+25}.

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

Cancel out x^{2}+5x+25 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and x+3 is \left(x-2\right)\left(x+3\right). Multiply \frac{x}{x-2} times \frac{x+3}{x+3}. Multiply \frac{2}{x+3} times \frac{x-2}{x-2}.

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

Since \frac{x\left(x+3\right)}{\left(x-2\right)\left(x+3\right)} and \frac{2\left(x-2\right)}{\left(x-2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in x\left(x+3\right)+2\left(x-2\right).

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

Combine like terms in x^{2}+3x+2x-4.

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

Factor x^{2}+x-6.

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

Since \frac{x^{2}+5x-4}{\left(x-2\right)\left(x+3\right)} and \frac{3}{\left(x-2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in x^{2}+5x-4-3.

$$\frac{x-5}{\frac{x^{2}+5x-7}{\left(x-2\right)\left(x+3\right)}}$$

Divide x-5 by \frac{x^{2}+5x-7}{\left(x-2\right)\left(x+3\right)} by multiplying x-5 by the reciprocal of \frac{x^{2}+5x-7}{\left(x-2\right)\left(x+3\right)}.

$$\frac{\left(x-5\right)\left(x-2\right)\left(x+3\right)}{x^{2}+5x-7}$$

Use the distributive property to multiply x-5 by x-2 and combine like terms.

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

Use the distributive property to multiply x^{2}-7x+10 by x+3 and combine like terms.

$$\frac{x^{3}-4x^{2}-11x+30}{x^{2}+5x-7}$$

Expand

$$\frac{x^{3}-4x^{2}-11x+30}{x^{2}+5x-7}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{3}-125}{x^{2}+5x+25}.

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

Cancel out x^{2}+5x+25 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and x+3 is \left(x-2\right)\left(x+3\right). Multiply \frac{x}{x-2} times \frac{x+3}{x+3}. Multiply \frac{2}{x+3} times \frac{x-2}{x-2}.

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

Since \frac{x\left(x+3\right)}{\left(x-2\right)\left(x+3\right)} and \frac{2\left(x-2\right)}{\left(x-2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in x\left(x+3\right)+2\left(x-2\right).

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

Combine like terms in x^{2}+3x+2x-4.

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

Factor x^{2}+x-6.

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

Since \frac{x^{2}+5x-4}{\left(x-2\right)\left(x+3\right)} and \frac{3}{\left(x-2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in x^{2}+5x-4-3.

$$\frac{x-5}{\frac{x^{2}+5x-7}{\left(x-2\right)\left(x+3\right)}}$$

Divide x-5 by \frac{x^{2}+5x-7}{\left(x-2\right)\left(x+3\right)} by multiplying x-5 by the reciprocal of \frac{x^{2}+5x-7}{\left(x-2\right)\left(x+3\right)}.

$$\frac{\left(x-5\right)\left(x-2\right)\left(x+3\right)}{x^{2}+5x-7}$$

Use the distributive property to multiply x-5 by x-2 and combine like terms.

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

Use the distributive property to multiply x^{2}-7x+10 by x+3 and combine like terms.

$$\frac{x^{3}-4x^{2}-11x+30}{x^{2}+5x-7}$$