Question

Add:$\frac{1}{m^2-m-2}+\frac{5m}{m^2+3m+2}$

Short answer

The simplification of the rational expression is$\frac{5m^2-9m+2}{\left(m+1\right)\left(m-2\right)\left(m+2\right)}$.

Step-by-step solution

Given information

The given rational expression is$\frac{1}{m^2-m-2}+\frac{5m}{m^2+3m+2}$

The simplification of the rational expression 

$$\begin{gathered} \frac{1}{m^2-2m+m-2}+\frac{5m}{m^2+2m+m+2} \\ \frac{1}{m\left(m-2\right)+1\left(m-2\right)}+\frac{5m}{m\left(m+2\right)+1\left(m+2\right)} \\ \frac{1}{\left(m+1\right)\left(m-2\right)}+\frac{5m}{\left(m+1\right)\left(m+2\right)} \end{gathered}$$

The LCD is $\left(m+1\right)\left(m-2\right)\left(m+2\right)$.

$$\begin{gathered} \frac{1\left(m+2\right)}{\left(m+1\right)\left(m-2\right)\left(m+2\right)}+\frac{5m\left(m-2\right)}{\left(m+1\right)\left(m-2\right)\left(m+2\right)} \\ \frac{m+2}{\left(m+1\right)\left(m-2\right)\left(m+2\right)}+\frac{5m^2-10m}{\left(m+1\right)\left(m-2\right)\left(m+2\right)} \\ \frac{m+2+5m^2-10m}{\left(m+1\right)\left(m-2\right)\left(m+2\right)} \\ \frac{5m^2-9m+2}{\left(m+1\right)\left(m-2\right)\left(m+2\right)} \end{gathered}$$