Question

$\frac{5}{n + 4}+\frac{5n}{2n - 6}$
$\frac{an^{2}+bn + c}{2(n - 3)(n + 4)}$
$a=$
$b=$
$c=$

Explanation:

Step1: Find common - denominator

The common denominator of \(n + 4\) and \(2n-6=2(n - 3)\) is \(2(n - 3)(n + 4)\). Rewrite the fractions: \(\frac{5}{n + 4}\times\frac{2(n - 3)}{2(n - 3)}=\frac{10(n - 3)}{2(n - 3)(n + 4)}\) and \(\frac{5n}{2n-6}\times\frac{n + 4}{n + 4}=\frac{5n(n + 4)}{2(n - 3)(n + 4)}\).

Step2: Add the fractions

\(\frac{10(n - 3)}{2(n - 3)(n + 4)}+\frac{5n(n + 4)}{2(n - 3)(n + 4)}=\frac{10(n - 3)+5n(n + 4)}{2(n - 3)(n + 4)}\).
Expand the numerator: \(10(n - 3)+5n(n + 4)=10n-30 + 5n^{2}+20n\).

Step3: Simplify the numerator

Combine like - terms: \(5n^{2}+(10n + 20n)-30=5n^{2}+30n-30\).
Since \(\frac{5n^{2}+30n-30}{2(n - 3)(n + 4)}=\frac{An^{2}+Bn + C}{2(n - 3)(n + 4)}\), then \(A = 5\), \(B=30\), \(C=-30\).

Answer:

A = 5
B = 30
C = - 30