Solve g(x)=x^3+5x/x^2-25divx^2+2x-24/x^2+x-30

$$g ( x ) = \frac { x ^ { 3 } + 5 x } { x ^ { 2 } - 25 } \div \frac { x ^ { 2 } + 2 x - 24 } { x ^ { 2 } + x - 30 }$$

Evaluate

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

Short Solution Steps

Divide \frac{x^{3}+5x}{x^{2}-25} by \frac{x^{2}+2x-24}{x^{2}+x-30} by multiplying \frac{x^{3}+5x}{x^{2}-25} by the reciprocal of \frac{x^{2}+2x-24}{x^{2}+x-30}.

$$\frac{\left(x^{3}+5x\right)\left(x^{2}+x-30\right)}{\left(x^{2}-25\right)\left(x^{2}+2x-24\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-5\right)\left(x+6\right) in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{3}+5x}{x^{2}+x-20}$$

Expand

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

Short Solution Steps

Divide \frac{x^{3}+5x}{x^{2}-25} by \frac{x^{2}+2x-24}{x^{2}+x-30} by multiplying \frac{x^{3}+5x}{x^{2}-25} by the reciprocal of \frac{x^{2}+2x-24}{x^{2}+x-30}.

$$\frac{\left(x^{3}+5x\right)\left(x^{2}+x-30\right)}{\left(x^{2}-25\right)\left(x^{2}+2x-24\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-5\right)\left(x+6\right) in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{3}+5x}{x^{2}+x-20}$$