# How do you write the partial fraction decomposition of the rational expression  ( x-2 ) / (x^2 + 4x + 3)?

Dec 16, 2015

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

#### Explanation:

Factor the denominator.

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

$x - 2 = A \left(x + 3\right) + B \left(x + 1\right)$

If $x = - 3$:

$- 3 - 2 = B \left(- 2\right)$
$B = \frac{5}{2}$

If $x = - 1$:

$- 1 - 2 = A \left(2\right)$
$A = - \frac{3}{2}$

Plug the values back in.

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