Solve x^2+2x+2+x+7/x^2-2

$$x ^ { 2 } + 2 x + 2 + \frac { x + 7 } { x ^ { 2 } - 2 }$$

Evaluate

$$\frac{x^{4}+2x^{3}-3x+3}{x^{2}-2}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2}+2x+2 times \frac{x^{2}-2}{x^{2}-2}.

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

Since \frac{\left(x^{2}+2x+2\right)\left(x^{2}-2\right)}{x^{2}-2} and \frac{x+7}{x^{2}-2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x^{2}+2x+2\right)\left(x^{2}-2\right)+x+7.

$$\frac{x^{4}-2x^{2}+2x^{3}-4x+2x^{2}-4+x+7}{x^{2}-2}$$

Combine like terms in x^{4}-2x^{2}+2x^{3}-4x+2x^{2}-4+x+7.

$$\frac{x^{4}+2x^{3}-3x+3}{x^{2}-2}$$

Expand

$$\frac{x^{4}+2x^{3}-3x+3}{x^{2}-2}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2}+2x+2 times \frac{x^{2}-2}{x^{2}-2}.

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

Since \frac{\left(x^{2}+2x+2\right)\left(x^{2}-2\right)}{x^{2}-2} and \frac{x+7}{x^{2}-2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x^{2}+2x+2\right)\left(x^{2}-2\right)+x+7.

$$\frac{x^{4}-2x^{2}+2x^{3}-4x+2x^{2}-4+x+7}{x^{2}-2}$$

Combine like terms in x^{4}-2x^{2}+2x^{3}-4x+2x^{2}-4+x+7.

$$\frac{x^{4}+2x^{3}-3x+3}{x^{2}-2}$$