# How do you combine (5x+2)/(x-4) + (x+3)/(x+1)?

May 20, 2015

Try this: May 21, 2015

The answer is $\frac{2 \left(3 {x}^{2} + 3 x - 5\right)}{\left(x + 1\right) \left(x - 4\right)}$.

Sorry, this is a long answer.

Problem: Combine $\frac{5 x + 2}{x - 4} + \frac{x + 3}{x + 1}$ .

Multiply the numerator and denominator of $\frac{5 x + 2}{x - 4}$ times $\left(x + 1\right)$.

$\frac{\left(5 x + 2\right) \left(x + 1\right)}{\left(x + 1\right) \left(x - 4\right)}$

Multiply the numerator and denominator of $\frac{x + 3}{x + 1}$ times $\left(x - 4\right)$.

$\frac{\left(x + 3\right) \left(x - 4\right)}{\left(x + 1\right) \left(x - 4\right)}$

We now have common denominators. Combine terms.

$\frac{\left(5 x + 2\right) \left(x + 1\right) + \left(x + 3\right) \left(x - 4\right)}{\left(x + 1\right) \left(x - 4\right)}$ =

(5x^2+5x+2x+2+x^2-4x+3x-12)/((x+1)(x-4) =

Group like terms.

$\frac{5 {x}^{2} + {x}^{2} + 5 x + 2 x - 4 x + 3 x + 2 - 12}{\left(x + 1\right) \left(x - 4\right)}$ =

Simplify.

$\frac{6 {x}^{2} + 6 x - 10}{\left(x + 1\right) \left(x - 4\right)}$

Factor out the GCF $2$.

$\frac{2 \left(3 {x}^{2} + 3 x - 5\right)}{\left(x + 1\right) \left(x - 4\right)}$