Solve frac{x^{2}-x}{x^2-2x-3}divx^2-4x+3/1-x^2+2x/x^2-9

$$\frac { x ^ { 2 } - x } { x ^ { 2 } - 2 x - 3 } \div \frac { x ^ { 2 } - 4 x + 3 } { 1 - x ^ { 2 } } + \frac { 2 x } { x ^ { 2 } - 9 }$$

Evaluate

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

Short Solution Steps

Divide \frac{x^{2}-x}{x^{2}-2x-3} by \frac{x^{2}-4x+3}{1-x^{2}} by multiplying \frac{x^{2}-x}{x^{2}-2x-3} by the reciprocal of \frac{x^{2}-4x+3}{1-x^{2}}.

$$\frac{\left(x^{2}-x\right)\left(1-x^{2}\right)}{\left(x^{2}-2x-3\right)\left(x^{2}-4x+3\right)}+\frac{2x}{x^{2}-9}$$

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

$$\frac{x\left(-x-1\right)\left(x-1\right)^{2}}{\left(x-1\right)\left(x+1\right)\left(x-3\right)^{2}}+\frac{2x}{x^{2}-9}$$

Extract the negative sign in -1-x.

$$\frac{-x\left(x+1\right)\left(x-1\right)^{2}}{\left(x-1\right)\left(x+1\right)\left(x-3\right)^{2}}+\frac{2x}{x^{2}-9}$$

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

$$\frac{-x\left(x-1\right)}{\left(x-3\right)^{2}}+\frac{2x}{x^{2}-9}$$

Factor x^{2}-9.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)^{2} and \left(x-3\right)\left(x+3\right) is \left(x+3\right)\left(x-3\right)^{2}. Multiply \frac{-x\left(x-1\right)}{\left(x-3\right)^{2}} times \frac{x+3}{x+3}. Multiply \frac{2x}{\left(x-3\right)\left(x+3\right)} times \frac{x-3}{x-3}.

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

Since \frac{-x\left(x-1\right)\left(x+3\right)}{\left(x+3\right)\left(x-3\right)^{2}} and \frac{2x\left(x-3\right)}{\left(x+3\right)\left(x-3\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -x\left(x-1\right)\left(x+3\right)+2x\left(x-3\right).

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

Combine like terms in -x^{3}-3x^{2}+x^{2}+3x+2x^{2}-6x.

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

Expand \left(x+3\right)\left(x-3\right)^{2}.

$$\frac{-x^{3}-3x}{x^{3}-3x^{2}-9x+27}$$

Expand

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

Short Solution Steps

Divide \frac{x^{2}-x}{x^{2}-2x-3} by \frac{x^{2}-4x+3}{1-x^{2}} by multiplying \frac{x^{2}-x}{x^{2}-2x-3} by the reciprocal of \frac{x^{2}-4x+3}{1-x^{2}}.

$$\frac{\left(x^{2}-x\right)\left(1-x^{2}\right)}{\left(x^{2}-2x-3\right)\left(x^{2}-4x+3\right)}+\frac{2x}{x^{2}-9}$$

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

$$\frac{x\left(-x-1\right)\left(x-1\right)^{2}}{\left(x-1\right)\left(x+1\right)\left(x-3\right)^{2}}+\frac{2x}{x^{2}-9}$$

Extract the negative sign in -1-x.

$$\frac{-x\left(x+1\right)\left(x-1\right)^{2}}{\left(x-1\right)\left(x+1\right)\left(x-3\right)^{2}}+\frac{2x}{x^{2}-9}$$

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

$$\frac{-x\left(x-1\right)}{\left(x-3\right)^{2}}+\frac{2x}{x^{2}-9}$$

Factor x^{2}-9.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)^{2} and \left(x-3\right)\left(x+3\right) is \left(x+3\right)\left(x-3\right)^{2}. Multiply \frac{-x\left(x-1\right)}{\left(x-3\right)^{2}} times \frac{x+3}{x+3}. Multiply \frac{2x}{\left(x-3\right)\left(x+3\right)} times \frac{x-3}{x-3}.

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

Since \frac{-x\left(x-1\right)\left(x+3\right)}{\left(x+3\right)\left(x-3\right)^{2}} and \frac{2x\left(x-3\right)}{\left(x+3\right)\left(x-3\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -x\left(x-1\right)\left(x+3\right)+2x\left(x-3\right).

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

Combine like terms in -x^{3}-3x^{2}+x^{2}+3x+2x^{2}-6x.

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

Expand \left(x+3\right)\left(x-3\right)^{2}.

$$\frac{-x^{3}-3x}{x^{3}-3x^{2}-9x+27}$$