Solve x^2/x^2+6x+9-4/x^2-9=

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

Evaluate

$$\frac{x^{3}-3x^{2}-4x-12}{\left(x-3\right)\left(x+3\right)^{2}}$$

Short Solution Steps

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)^{2} and \left(x-3\right)\left(x+3\right) is \left(x-3\right)\left(x+3\right)^{2}. Multiply \frac{x^{2}}{\left(x+3\right)^{2}} times \frac{x-3}{x-3}. Multiply \frac{4}{\left(x-3\right)\left(x+3\right)} times \frac{x+3}{x+3}.

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

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

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

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

$$\frac{x^{3}-3x^{2}-4x-12}{\left(x-3\right)\left(x+3\right)^{2}}$$

Expand \left(x-3\right)\left(x+3\right)^{2}.

$$\frac{x^{3}-3x^{2}-4x-12}{x^{3}+3x^{2}-9x-27}$$

Differentiate w.r.t. x

$$\frac{2x\left(3x^{2}-14x+39\right)}{\left(x-3\right)^{2}\left(x+3\right)^{3}}$$