Solve x^2-2x-16/x^2+x*x^2+7x/x^2+12x+27

$$\frac { x ^ { 2 } - 2 x - 16 } { x ^ { 2 } + x } \cdot \frac { x ^ { 2 } + 7 x } { x ^ { 2 } + 12 x + 27 }$$

Evaluate

$$\frac{\left(x+7\right)\left(x^{2}-2x-16\right)}{\left(x+1\right)\left(x^{2}+12x+27\right)}$$

Short Solution Steps

Multiply \frac{x^{2}-2x-16}{x^{2}+x} times \frac{x^{2}+7x}{x^{2}+12x+27} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x^{2}-2x-16\right)\left(x^{2}+7x\right)}{\left(x^{2}+x\right)\left(x^{2}+12x+27\right)}$$

Factor the expressions that are not already factored.

$$\frac{x\left(x+7\right)\left(x-\left(\sqrt{17}+1\right)\right)\left(x-\left(-\sqrt{17}+1\right)\right)}{x\left(x+1\right)\left(x+3\right)\left(x+9\right)}$$

Cancel out x in both numerator and denominator.

$$\frac{\left(x+7\right)\left(x-\left(\sqrt{17}+1\right)\right)\left(x-\left(-\sqrt{17}+1\right)\right)}{\left(x+1\right)\left(x+3\right)\left(x+9\right)}$$

Expand the expression.

$$\frac{x^{3}+5x^{2}-30x-112}{x^{3}+13x^{2}+39x+27}$$

Expand

$$\frac{x^{3}+5x^{2}-30x-112}{\left(x+1\right)\left(x^{2}+12x+27\right)}$$

Short Solution Steps

Multiply \frac{x^{2}-2x-16}{x^{2}+x} times \frac{x^{2}+7x}{x^{2}+12x+27} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x^{2}-2x-16\right)\left(x^{2}+7x\right)}{\left(x^{2}+x\right)\left(x^{2}+12x+27\right)}$$

Factor the expressions that are not already factored.

$$\frac{x\left(x+7\right)\left(x-\left(\sqrt{17}+1\right)\right)\left(x-\left(-\sqrt{17}+1\right)\right)}{x\left(x+1\right)\left(x+3\right)\left(x+9\right)}$$

Cancel out x in both numerator and denominator.

$$\frac{\left(x+7\right)\left(x-\left(\sqrt{17}+1\right)\right)\left(x-\left(-\sqrt{17}+1\right)\right)}{\left(x+1\right)\left(x+3\right)\left(x+9\right)}$$

Expand the expression.

$$\frac{x^{3}+5x^{2}-30x-112}{x^{3}+13x^{2}+39x+27}$$