Solve 8/x+2+20/x+6/frac{7x+22{x^2+8x+12}}

$$\frac { \frac { 8 } { x + 2 } + \frac { 20 } { x + 6 } } { \frac { 7 x + 22 } { x ^ { 2 } + 8 x + 12 } }$$

Evaluate

$$4$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and x+6 is \left(x+2\right)\left(x+6\right). Multiply \frac{8}{x+2} times \frac{x+6}{x+6}. Multiply \frac{20}{x+6} times \frac{x+2}{x+2}.

$$\frac{\frac{8\left(x+6\right)}{\left(x+2\right)\left(x+6\right)}+\frac{20\left(x+2\right)}{\left(x+2\right)\left(x+6\right)}}{\frac{7x+22}{x^{2}+8x+12}}$$

Since \frac{8\left(x+6\right)}{\left(x+2\right)\left(x+6\right)} and \frac{20\left(x+2\right)}{\left(x+2\right)\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{8\left(x+6\right)+20\left(x+2\right)}{\left(x+2\right)\left(x+6\right)}}{\frac{7x+22}{x^{2}+8x+12}}$$

Do the multiplications in 8\left(x+6\right)+20\left(x+2\right).

$$\frac{\frac{8x+48+20x+40}{\left(x+2\right)\left(x+6\right)}}{\frac{7x+22}{x^{2}+8x+12}}$$

Combine like terms in 8x+48+20x+40.

$$\frac{\frac{28x+88}{\left(x+2\right)\left(x+6\right)}}{\frac{7x+22}{x^{2}+8x+12}}$$

Divide \frac{28x+88}{\left(x+2\right)\left(x+6\right)} by \frac{7x+22}{x^{2}+8x+12} by multiplying \frac{28x+88}{\left(x+2\right)\left(x+6\right)} by the reciprocal of \frac{7x+22}{x^{2}+8x+12}.

$$\frac{\left(28x+88\right)\left(x^{2}+8x+12\right)}{\left(x+2\right)\left(x+6\right)\left(7x+22\right)}$$

Factor the expressions that are not already factored.

$$\frac{4\left(x+2\right)\left(x+6\right)\left(7x+22\right)}{\left(x+2\right)\left(x+6\right)\left(7x+22\right)}$$

Cancel out \left(x+2\right)\left(x+6\right)\left(7x+22\right) in both numerator and denominator.

$$4$$

Factor

$$2^{2}$$