Solve x^2+x/x^2-25+x^2-1/x^2+11x+30

$$\frac { x ^ { 2 } + x } { x ^ { 2 } - 25 } + \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 11 x + 30 }$$

Evaluate

$$\frac{\left(x+1\right)\left(2x^{2}+5\right)}{\left(x+6\right)\left(x^{2}-25\right)}$$

Short Solution Steps

Factor x^{2}-25. Factor x^{2}+11x+30.

$$\frac{x^{2}+x}{\left(x-5\right)\left(x+5\right)}+\frac{x^{2}-1}{\left(x+5\right)\left(x+6\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x+5\right) and \left(x+5\right)\left(x+6\right) is \left(x-5\right)\left(x+5\right)\left(x+6\right). Multiply \frac{x^{2}+x}{\left(x-5\right)\left(x+5\right)} times \frac{x+6}{x+6}. Multiply \frac{x^{2}-1}{\left(x+5\right)\left(x+6\right)} times \frac{x-5}{x-5}.

$$\frac{\left(x^{2}+x\right)\left(x+6\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}+\frac{\left(x^{2}-1\right)\left(x-5\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Since \frac{\left(x^{2}+x\right)\left(x+6\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)} and \frac{\left(x^{2}-1\right)\left(x-5\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x^{2}+x\right)\left(x+6\right)+\left(x^{2}-1\right)\left(x-5\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Do the multiplications in \left(x^{2}+x\right)\left(x+6\right)+\left(x^{2}-1\right)\left(x-5\right).

$$\frac{x^{3}+6x^{2}+x^{2}+6x+x^{3}-5x^{2}-x+5}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Combine like terms in x^{3}+6x^{2}+x^{2}+6x+x^{3}-5x^{2}-x+5.

$$\frac{2x^{3}+2x^{2}+5x+5}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Expand \left(x-5\right)\left(x+5\right)\left(x+6\right).

$$\frac{2x^{3}+2x^{2}+5x+5}{x^{3}+6x^{2}-25x-150}$$

Expand

$$\frac{2x^{3}+2x^{2}+5x+5}{\left(x+6\right)\left(x^{2}-25\right)}$$

Short Solution Steps

Factor x^{2}-25. Factor x^{2}+11x+30.

$$\frac{x^{2}+x}{\left(x-5\right)\left(x+5\right)}+\frac{x^{2}-1}{\left(x+5\right)\left(x+6\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x+5\right) and \left(x+5\right)\left(x+6\right) is \left(x-5\right)\left(x+5\right)\left(x+6\right). Multiply \frac{x^{2}+x}{\left(x-5\right)\left(x+5\right)} times \frac{x+6}{x+6}. Multiply \frac{x^{2}-1}{\left(x+5\right)\left(x+6\right)} times \frac{x-5}{x-5}.

$$\frac{\left(x^{2}+x\right)\left(x+6\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}+\frac{\left(x^{2}-1\right)\left(x-5\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Since \frac{\left(x^{2}+x\right)\left(x+6\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)} and \frac{\left(x^{2}-1\right)\left(x-5\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x^{2}+x\right)\left(x+6\right)+\left(x^{2}-1\right)\left(x-5\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Do the multiplications in \left(x^{2}+x\right)\left(x+6\right)+\left(x^{2}-1\right)\left(x-5\right).

$$\frac{x^{3}+6x^{2}+x^{2}+6x+x^{3}-5x^{2}-x+5}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Combine like terms in x^{3}+6x^{2}+x^{2}+6x+x^{3}-5x^{2}-x+5.

$$\frac{2x^{3}+2x^{2}+5x+5}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Expand \left(x-5\right)\left(x+5\right)\left(x+6\right).

$$\frac{2x^{3}+2x^{2}+5x+5}{x^{3}+6x^{2}-25x-150}$$