Solve x^2-4x+4/x^2+6x+9+1/frac{x-2{x+3}}

$$\frac{ \frac{ { x }^{ 2 } -4x+4 }{ { x }^{ 2 } +6x+9 } +1 }{ \frac{ x-2 }{ x+3 } }$$

Evaluate

$$\frac{2x^{2}+2x+13}{\left(x-2\right)\left(x+3\right)}$$

Short Solution Steps

Factor x^{2}+6x+9.

$$\frac{\frac{x^{2}-4x+4}{\left(x+3\right)^{2}}+1}{\frac{x-2}{x+3}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(x+3\right)^{2}}{\left(x+3\right)^{2}}.

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

Since \frac{x^{2}-4x+4}{\left(x+3\right)^{2}} and \frac{\left(x+3\right)^{2}}{\left(x+3\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in x^{2}-4x+4+\left(x+3\right)^{2}.

$$\frac{\frac{x^{2}-4x+4+x^{2}+6x+9}{\left(x+3\right)^{2}}}{\frac{x-2}{x+3}}$$

Combine like terms in x^{2}-4x+4+x^{2}+6x+9.

$$\frac{\frac{2x^{2}+2x+13}{\left(x+3\right)^{2}}}{\frac{x-2}{x+3}}$$

Divide \frac{2x^{2}+2x+13}{\left(x+3\right)^{2}} by \frac{x-2}{x+3} by multiplying \frac{2x^{2}+2x+13}{\left(x+3\right)^{2}} by the reciprocal of \frac{x-2}{x+3}.

$$\frac{\left(2x^{2}+2x+13\right)\left(x+3\right)}{\left(x+3\right)^{2}\left(x-2\right)}$$

Cancel out x+3 in both numerator and denominator.

$$\frac{2x^{2}+2x+13}{\left(x-2\right)\left(x+3\right)}$$

Use the distributive property to multiply x-2 by x+3 and combine like terms.

$$\frac{2x^{2}+2x+13}{x^{2}+x-6}$$

Expand

$$\frac{2x^{2}+2x+13}{\left(x-2\right)\left(x+3\right)}$$

Short Solution Steps

Factor x^{2}+6x+9.

$$\frac{\frac{x^{2}-4x+4}{\left(x+3\right)^{2}}+1}{\frac{x-2}{x+3}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(x+3\right)^{2}}{\left(x+3\right)^{2}}.

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

Since \frac{x^{2}-4x+4}{\left(x+3\right)^{2}} and \frac{\left(x+3\right)^{2}}{\left(x+3\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in x^{2}-4x+4+\left(x+3\right)^{2}.

$$\frac{\frac{x^{2}-4x+4+x^{2}+6x+9}{\left(x+3\right)^{2}}}{\frac{x-2}{x+3}}$$

Combine like terms in x^{2}-4x+4+x^{2}+6x+9.

$$\frac{\frac{2x^{2}+2x+13}{\left(x+3\right)^{2}}}{\frac{x-2}{x+3}}$$

Divide \frac{2x^{2}+2x+13}{\left(x+3\right)^{2}} by \frac{x-2}{x+3} by multiplying \frac{2x^{2}+2x+13}{\left(x+3\right)^{2}} by the reciprocal of \frac{x-2}{x+3}.

$$\frac{\left(2x^{2}+2x+13\right)\left(x+3\right)}{\left(x+3\right)^{2}\left(x-2\right)}$$

Cancel out x+3 in both numerator and denominator.

$$\frac{2x^{2}+2x+13}{\left(x-2\right)\left(x+3\right)}$$

Use the distributive property to multiply x-2 by x+3 and combine like terms.

$$\frac{2x^{2}+2x+13}{x^{2}+x-6}$$