Question

$\frac{5}{2x + 4}-\frac{4x}{x - 1}$
$\frac{ax^{2}+bx + c}{2(x - 1)(x + 2)}$
$a=$
$b=$
$c=$

Explanation:

Step1: Find a common - denominator

First, factor \(2x + 4=2(x + 2)\). The common denominator of \(\frac{5}{2x + 4}\) and \(\frac{4x}{x - 1}\) is \(2(x - 1)(x + 2)\). Rewrite the fractions: \(\frac{5}{2(x + 2)}-\frac{4x}{x - 1}=\frac{5(x - 1)-4x\times2(x + 2)}{2(x - 1)(x + 2)}\).

Step2: Expand the numerator

Expand \(5(x - 1)-4x\times2(x + 2)\):
\[

$$\begin{align*} 5(x - 1)-8x(x + 2)&=5x-5-(8x^{2}+16x)\\ &=5x - 5-8x^{2}-16x\\ &=-8x^{2}-11x - 5 \end{align*}$$

\]
Since \(\frac{-8x^{2}-11x - 5}{2(x - 1)(x + 2)}=\frac{Ax^{2}+Bx + C}{2(x - 1)(x + 2)}\), then \(Ax^{2}+Bx + C=-8x^{2}-11x - 5\).

Answer:

A = - 8
B = - 11
C = - 5