Solve frac{x^{3}-1}{x^{2}-2x^{2}-3x}*x+1/x^2+x-2divx^2+x+1/x^3-x^2-6x

$$\frac { x ^ { 3 } - 1 } { x ^ { 2 } - 2 x ^ { 2 } - 3 x } \cdot \frac { x + 1 } { x ^ { 2 } + x - 2 } \div \frac { x ^ { 2 } + x + 1 } { x ^ { 3 } - x ^ { 2 } - 6 x }$$

Evaluate

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

Short Solution Steps

Combine x^{2} and -2x^{2} to get -x^{2}.

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

Multiply \frac{x^{3}-1}{-x^{2}-3x} times \frac{x+1}{x^{2}+x-2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(x^{3}-1\right)\left(x+1\right)}{\left(-x^{2}-3x\right)\left(x^{2}+x-2\right)}}{\frac{x^{2}+x+1}{x^{3}-x^{2}-6x}}$$

Divide \frac{\left(x^{3}-1\right)\left(x+1\right)}{\left(-x^{2}-3x\right)\left(x^{2}+x-2\right)} by \frac{x^{2}+x+1}{x^{3}-x^{2}-6x} by multiplying \frac{\left(x^{3}-1\right)\left(x+1\right)}{\left(-x^{2}-3x\right)\left(x^{2}+x-2\right)} by the reciprocal of \frac{x^{2}+x+1}{x^{3}-x^{2}-6x}.

$$\frac{\left(x^{3}-1\right)\left(x+1\right)\left(x^{3}-x^{2}-6x\right)}{\left(-x^{2}-3x\right)\left(x^{2}+x-2\right)\left(x^{2}+x+1\right)}$$

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

Expand

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

Short Solution Steps

Combine x^{2} and -2x^{2} to get -x^{2}.

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

Multiply \frac{x^{3}-1}{-x^{2}-3x} times \frac{x+1}{x^{2}+x-2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(x^{3}-1\right)\left(x+1\right)}{\left(-x^{2}-3x\right)\left(x^{2}+x-2\right)}}{\frac{x^{2}+x+1}{x^{3}-x^{2}-6x}}$$

Divide \frac{\left(x^{3}-1\right)\left(x+1\right)}{\left(-x^{2}-3x\right)\left(x^{2}+x-2\right)} by \frac{x^{2}+x+1}{x^{3}-x^{2}-6x} by multiplying \frac{\left(x^{3}-1\right)\left(x+1\right)}{\left(-x^{2}-3x\right)\left(x^{2}+x-2\right)} by the reciprocal of \frac{x^{2}+x+1}{x^{3}-x^{2}-6x}.

$$\frac{\left(x^{3}-1\right)\left(x+1\right)\left(x^{3}-x^{2}-6x\right)}{\left(-x^{2}-3x\right)\left(x^{2}+x-2\right)\left(x^{2}+x+1\right)}$$

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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