# How do you express ( x-2 ) / (x^2 + 4x + 3) in partial fractions?

Feb 6, 2016

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

#### Explanation:

The first step here is to factor the denominator

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

since these factors are linear the numerator will be a constant

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

the next step is to multiply both sides by (x+1)(x+3)

hence x - 2 = A(x+3) + B(x+1)

Note that if x = - 3 and x = -1 then the terms with A and B will be zero

let x = - 3 : - 5 = -2B → B$= \frac{5}{2}$

let x = - 1 : - 3 = 2A → A #= -3/2

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