Solve 8/x+4+12/x+5/frac{5x+22{x^2+9x+20}}

$$\frac { \frac { 8 } { x + 4 } + \frac { 12 } { x + 5 } } { \frac { 5 x + 22 } { x ^ { 2 } + 9 x + 20 } }$$

Evaluate

$$4$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+4 and x+5 is \left(x+4\right)\left(x+5\right). Multiply \frac{8}{x+4} times \frac{x+5}{x+5}. Multiply \frac{12}{x+5} times \frac{x+4}{x+4}.

$$\frac{\frac{8\left(x+5\right)}{\left(x+4\right)\left(x+5\right)}+\frac{12\left(x+4\right)}{\left(x+4\right)\left(x+5\right)}}{\frac{5x+22}{x^{2}+9x+20}}$$

Since \frac{8\left(x+5\right)}{\left(x+4\right)\left(x+5\right)} and \frac{12\left(x+4\right)}{\left(x+4\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{8\left(x+5\right)+12\left(x+4\right)}{\left(x+4\right)\left(x+5\right)}}{\frac{5x+22}{x^{2}+9x+20}}$$

Do the multiplications in 8\left(x+5\right)+12\left(x+4\right).

$$\frac{\frac{8x+40+12x+48}{\left(x+4\right)\left(x+5\right)}}{\frac{5x+22}{x^{2}+9x+20}}$$

Combine like terms in 8x+40+12x+48.

$$\frac{\frac{20x+88}{\left(x+4\right)\left(x+5\right)}}{\frac{5x+22}{x^{2}+9x+20}}$$

Divide \frac{20x+88}{\left(x+4\right)\left(x+5\right)} by \frac{5x+22}{x^{2}+9x+20} by multiplying \frac{20x+88}{\left(x+4\right)\left(x+5\right)} by the reciprocal of \frac{5x+22}{x^{2}+9x+20}.

$$\frac{\left(20x+88\right)\left(x^{2}+9x+20\right)}{\left(x+4\right)\left(x+5\right)\left(5x+22\right)}$$

Factor the expressions that are not already factored.

$$\frac{4\left(x+4\right)\left(x+5\right)\left(5x+22\right)}{\left(x+4\right)\left(x+5\right)\left(5x+22\right)}$$

Cancel out \left(x+4\right)\left(x+5\right)\left(5x+22\right) in both numerator and denominator.

$$4$$

Factor

$$2^{2}$$