Solve x^2-2x+8/x^2+4x+4-2/x+2

$$\frac{ { x }^{ 2 } -2x+8 }{ { x }^{ 2 } +4x+4 } - \frac{ 2 }{ x+2 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}+4x+4.

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

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

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

Since \frac{x^{2}-2x+8}{\left(x+2\right)^{2}} and \frac{2\left(x+2\right)}{\left(x+2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x^{2}-2x+8-2\left(x+2\right).

$$\frac{x^{2}-2x+8-2x-4}{\left(x+2\right)^{2}}$$

Combine like terms in x^{2}-2x+8-2x-4.

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

Expand \left(x+2\right)^{2}.

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

Expand

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

Short Solution Steps

Factor x^{2}+4x+4.

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

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

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

Since \frac{x^{2}-2x+8}{\left(x+2\right)^{2}} and \frac{2\left(x+2\right)}{\left(x+2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x^{2}-2x+8-2\left(x+2\right).

$$\frac{x^{2}-2x+8-2x-4}{\left(x+2\right)^{2}}$$

Combine like terms in x^{2}-2x+8-2x-4.

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

Expand \left(x+2\right)^{2}.

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