Question

Simplify the complex rational expression by using the LCD:

$\frac{\frac{x^2}{x^2+7x+10}}{\frac{4}{x+2}+\frac{1}{x+5}}$

Short answer

The simplified expression is$\frac{x^2}{5x+22}$.

Step-by-step solution

Step 1. Given Information. 

We have:

$\frac{\frac{x^2}{x^2+7x+10}}{\frac{4}{x+2}+\frac{1}{x+5}}$

Step 2. Find the LCD of all fractions in complex rational expression.

The least common divisor is $\left(x+2\right)\left(x+5\right).$

Multiply the numerator and denominator by the LCD:

$=\frac{\left(x+2\right)(x+5)\left(\frac{x^2}{x^2+7x+10}\right)}{\left(x+2\right)(x+5)\left(\frac{4}{x+2}+\frac{1}{x+5}\right)}$

Step 3. Simplify the expression. 

Further, calculate:

$$\begin{gathered} =\frac{\left(x+2\right)(x+5)\left(\frac{x^2}{x^2+5x+2x+10}\right)}{\left(x+2\right)(x+5)\left(\frac{4\left(x+5\right)+1\left(x+2\right)}{\left(x+2\right)\left(x+5\right)}\right)} \\ =\frac{\left(x+2\right)(x+5)\left(\frac{x^2}{x(x+5)+2(x+5)}\right)}{\left(x+2\right)(x+5)\left(\frac{4\left(x+5\right)+1\left(x+2\right)}{\left(x+2\right)\left(x+5\right)}\right)} \\ =\frac{\bcancel{\left(x+2\right)(x+5)}\left(\frac{x^2}{\bcancel{(x+5)\left(x+2\right)}}\right)}{\bcancel{\left(x+2\right)(x+5)}\left(\frac{4\left(x+5\right)+1\left(x+2\right)}{\bcancel{\left(x+2\right)\left(x+5\right)}}\right)} \\ =\frac{x^2}{4x+20+x+2} \\ =\frac{x^2}{5x+22} \end{gathered}$$