Solve x^2+4x/x^2+15x+54-4/x+6

$$\frac { x ^ { 2 } + 4 x } { x ^ { 2 } + 15 x + 54 } - \frac { 4 } { x + 6 }$$

Evaluate

$$\frac{x-6}{x+9}$$

Short Solution Steps

Factor x^{2}+15x+54.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+6\right)\left(x+9\right) and x+6 is \left(x+6\right)\left(x+9\right). Multiply \frac{4}{x+6} times \frac{x+9}{x+9}.

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

Since \frac{x^{2}+4x}{\left(x+6\right)\left(x+9\right)} and \frac{4\left(x+9\right)}{\left(x+6\right)\left(x+9\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x^{2}+4x-4\left(x+9\right).

$$\frac{x^{2}+4x-4x-36}{\left(x+6\right)\left(x+9\right)}$$

Combine like terms in x^{2}+4x-4x-36.

$$\frac{x^{2}-36}{\left(x+6\right)\left(x+9\right)}$$

Factor the expressions that are not already factored in \frac{x^{2}-36}{\left(x+6\right)\left(x+9\right)}.

$$\frac{\left(x-6\right)\left(x+6\right)}{\left(x+6\right)\left(x+9\right)}$$

Cancel out x+6 in both numerator and denominator.

$$\frac{x-6}{x+9}$$

Expand

$$\frac{x-6}{x+9}$$

Short Solution Steps

Factor x^{2}+15x+54.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+6\right)\left(x+9\right) and x+6 is \left(x+6\right)\left(x+9\right). Multiply \frac{4}{x+6} times \frac{x+9}{x+9}.

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

Since \frac{x^{2}+4x}{\left(x+6\right)\left(x+9\right)} and \frac{4\left(x+9\right)}{\left(x+6\right)\left(x+9\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x^{2}+4x-4\left(x+9\right).

$$\frac{x^{2}+4x-4x-36}{\left(x+6\right)\left(x+9\right)}$$

Combine like terms in x^{2}+4x-4x-36.

$$\frac{x^{2}-36}{\left(x+6\right)\left(x+9\right)}$$

Factor the expressions that are not already factored in \frac{x^{2}-36}{\left(x+6\right)\left(x+9\right)}.

$$\frac{\left(x-6\right)\left(x+6\right)}{\left(x+6\right)\left(x+9\right)}$$

Cancel out x+6 in both numerator and denominator.

$$\frac{x-6}{x+9}$$