# How do you add or subtract (2)/(x^2 + 8x + 15) + (1)/(x^2 + 11x + 30)?

May 12, 2018

Answer is 3/((x+3)(x+6) OR $\frac{3}{{x}^{2} + 9 x + 18}$

#### Explanation:

First we simplify and find the factors of ${x}^{2} + 8 x + 15$ and ${x}^{2} + 11 x + 30$.

Step 1: Factors of ${x}^{2} + 8 x + 15$ equation
${x}^{2} + 8 x + 15$ -----> factors are 3 and 5 ($3 + 5 = 8$ and $3 \times 5 = 15$)
${x}^{2} + 8 x + 15$
${x}^{2} + 3 x + 5 x + 15$
$x \left(x + 3\right) + 5 \left(x + 3\right)$
$\left(\textcolor{red}{x + 3}\right) \left(\textcolor{red}{x + 5}\right)$

Step 2: Factors of ${x}^{2} + 11 x + 30$.
${x}^{2} + 11 x + 30$ ----> factors are 5 and 6 ($5 + 6 = 11$ and 5xx6=30)
${x}^{2} + 11 x + 30$
${x}^{2} + 6 x + 5 x + 30$
$x \left(x + 6\right) + 5 \left(x + 6\right)$
$\left(\textcolor{red}{x + 6}\right) \left(\textcolor{red}{x + 5}\right)$

Step 3: Re-write the original equation with the above common factors and simplify further

$\frac{2}{\left(x + 3\right) \left(x + 5\right)}$ + $\frac{1}{\left(x + 6\right) \left(x + 5\right)}$

Least Common Multiple is $\left(x + 3\right) \left(x + 6\right) \left(x + 5\right)$

$\frac{2 \left(x + 6\right) + 1 \left(x + 3\right)}{\left(x + 3\right) \left(x + 6\right) \left(x + 5\right)}$

$\frac{2 x + 12 + x + 3}{\left(x + 3\right) \left(x + 6\right) \left(x + 5\right)}$

$\frac{3 x + 15}{\left(x + 3\right) \left(x + 6\right) \left(x + 5\right)}$-----> simplify $3 x + 15$

$\frac{3 \left(x + 5\right)}{\left(x + 3\right) \left(x + 6\right) \left(x + 5\right)}$

$\frac{3 \cancel{\left(x + 5\right)}}{\left(x + 3\right) \left(x + 6\right) \cancel{x + 5}}$

3/((x+3)(x+6) OR $\frac{3}{{x}^{2} + 9 x + 18}$