Solve 2/x+5+4/x+3/frac{3x+13{x^2+8x+15}}

$$\frac { \frac { 2 } { x + 5 } + \frac { 4 } { x + 3 } } { \frac { 3 x + 13 } { x ^ { 2 } + 8 x + 15 } }$$

Evaluate

$$2$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+5 and x+3 is \left(x+3\right)\left(x+5\right). Multiply \frac{2}{x+5} times \frac{x+3}{x+3}. Multiply \frac{4}{x+3} times \frac{x+5}{x+5}.

$$\frac{\frac{2\left(x+3\right)}{\left(x+3\right)\left(x+5\right)}+\frac{4\left(x+5\right)}{\left(x+3\right)\left(x+5\right)}}{\frac{3x+13}{x^{2}+8x+15}}$$

Since \frac{2\left(x+3\right)}{\left(x+3\right)\left(x+5\right)} and \frac{4\left(x+5\right)}{\left(x+3\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{2\left(x+3\right)+4\left(x+5\right)}{\left(x+3\right)\left(x+5\right)}}{\frac{3x+13}{x^{2}+8x+15}}$$

Do the multiplications in 2\left(x+3\right)+4\left(x+5\right).

$$\frac{\frac{2x+6+4x+20}{\left(x+3\right)\left(x+5\right)}}{\frac{3x+13}{x^{2}+8x+15}}$$

Combine like terms in 2x+6+4x+20.

$$\frac{\frac{6x+26}{\left(x+3\right)\left(x+5\right)}}{\frac{3x+13}{x^{2}+8x+15}}$$

Divide \frac{6x+26}{\left(x+3\right)\left(x+5\right)} by \frac{3x+13}{x^{2}+8x+15} by multiplying \frac{6x+26}{\left(x+3\right)\left(x+5\right)} by the reciprocal of \frac{3x+13}{x^{2}+8x+15}.

$$\frac{\left(6x+26\right)\left(x^{2}+8x+15\right)}{\left(x+3\right)\left(x+5\right)\left(3x+13\right)}$$

Factor the expressions that are not already factored.

$$\frac{2\left(x+3\right)\left(x+5\right)\left(3x+13\right)}{\left(x+3\right)\left(x+5\right)\left(3x+13\right)}$$

Cancel out \left(x+3\right)\left(x+5\right)\left(3x+13\right) in both numerator and denominator.

$$2$$

Factor

$$2$$