Solve g(x)=x^3+2x/x^2-4divx^2+3x-40/x^2+6x-16

$$g ( x ) = \frac { x ^ { 3 } + 2 x } { x ^ { 2 } - 4 } \div \frac { x ^ { 2 } + 3 x - 40 } { x ^ { 2 } + 6 x - 16 }$$

Evaluate

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

Short Solution Steps

Divide \frac{x^{3}+2x}{x^{2}-4} by \frac{x^{2}+3x-40}{x^{2}+6x-16} by multiplying \frac{x^{3}+2x}{x^{2}-4} by the reciprocal of \frac{x^{2}+3x-40}{x^{2}+6x-16}.

$$\frac{\left(x^{3}+2x\right)\left(x^{2}+6x-16\right)}{\left(x^{2}-4\right)\left(x^{2}+3x-40\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-2\right)\left(x+8\right) in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{3}+2x}{x^{2}-3x-10}$$

Expand

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

Short Solution Steps

Divide \frac{x^{3}+2x}{x^{2}-4} by \frac{x^{2}+3x-40}{x^{2}+6x-16} by multiplying \frac{x^{3}+2x}{x^{2}-4} by the reciprocal of \frac{x^{2}+3x-40}{x^{2}+6x-16}.

$$\frac{\left(x^{3}+2x\right)\left(x^{2}+6x-16\right)}{\left(x^{2}-4\right)\left(x^{2}+3x-40\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-2\right)\left(x+8\right) in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{3}+2x}{x^{2}-3x-10}$$