Question

In the following exercise, find $R\left(x\right)=f\left(x\right)+g\left(x\right)$where $f\left(x\right)$and $g\left(x\right)$are given role="math" localid="1646707453949" $f\left(x\right)=\frac{2x^2+12x-11}{x^2+3x-10}$,role="math" localid="1646707470103" $g\left(x\right)=\frac{x+1}{2-x}$

Short answer

The solution is$\frac{x+8}{x+5}$

Step-by-step solution

Step 1. Given Information

$f\left(x\right)=\frac{2x^2+12x-11}{x^2+3x-10}$and$g\left(x\right)=\frac{x+1}{2-x}$

Step 2. Find Rx=fx+gx

The LCD of $x^2+3x-10,2-x:\left(x-2\right)\left(x+5\right)\left(-x+2\right)$

Rewrite each rational expression as an equivalent rational expression with LCD.

$$\begin{gathered} R\left(x\right)=f\left(x\right)+g\left(x\right) \\ R\left(x\right)=\frac{\left(2x^2+12x-11\right)\left(-x+2\right)}{\left(x-2\right)\left(x+5\right)\left(-x+2\right)}+\frac{-\left(x+1\right)\left(x+5\right)\left(-x+2\right)}{\left(x-2\right)\left(x+5\right)\left(-x+2\right)} \\ R\left(x\right)=\frac{\left(2x^2+12x-11\right)\left(-x+2\right)-\left(x+1\right)\left(x+5\right)\left(-x+2\right)}{\left(x-2\right)\left(x+5\right)\left(-x+2\right)} \\ R\left(x\right)=\frac{x+8}{x+5} \end{gathered}$$