Question

In the following exercises, perform the indicated operations

$\frac{2x}{x^2+10x+24}+\frac{3x}{x^2+8x+16}$

Short answer

Simplified expression is $\frac{5x^2+26x}{{(x+4)}^2(x+6)}$

Step-by-step solution

Step 1. Given information

Expression:

$\frac{2x}{x^2+10x+24}+\frac{3x}{x^2+8x+16}$

Step 2. Find LCD    

Denominator of first fraction is$x^2+10x+24$

Factorize this denominator

$$\begin{gathered} x^2+6x+4x+24 \\ =x(x+6)+4(x+6) \\ =(x+6)(x+4) \end{gathered}$$

Denominator of the second fraction is$x^2+8x+16$

Factorize this denominator

$$\begin{gathered} x^2+2.4.x+4^2 \\ ={(x+4)}^2 \end{gathered}$$

here $\left(x+4\right)$is common between both denominator

So the LCD is${(x+4)}^2(x+6)$

Step 3. Rewrite each rational expression as  an equivalent rational expression with the LCD  

$\frac{2x(x+4)}{{(x+4)}^2(x+6)}+\frac{3x(x+6)}{{(x+4)}^2(x+6)}$

Step 4.Simplify the numerators 

$\frac{2x^2+8x}{{(x+4)}^2(x+6)}+\frac{3x^2+18x}{{(x+4)}^2(x+6)}$

Step 5. Add numerators of the fractions    

$\frac{2x^2+8x+3x^2+18x}{{(x+4)}^2(x+6)}$

Step 6. Simplify numerator  

$\frac{5x^2+26x}{{(x+4)}^2(x+6)}$