Question

1. \\(\frac{6p}{2p + 2} + \frac{5}{p - 3}\\)

Explanation:

Step1: Factor the denominator

Factor \(2p + 2\) as \(2(p + 1)\). So the expression becomes \(\frac{6p}{2(p + 1)}+\frac{5}{p - 3}\). Simplify \(\frac{6p}{2(p + 1)}\) to \(\frac{3p}{p + 1}\). Now we have \(\frac{3p}{p + 1}+\frac{5}{p - 3}\).

Step2: Find the common denominator

The common denominator of \((p + 1)\) and \((p - 3)\) is \((p + 1)(p - 3)\).

Step3: Rewrite each fraction with the common denominator

For \(\frac{3p}{p + 1}\), multiply numerator and denominator by \((p - 3)\): \(\frac{3p(p - 3)}{(p + 1)(p - 3)}\). For \(\frac{5}{p - 3}\), multiply numerator and denominator by \((p + 1)\): \(\frac{5(p + 1)}{(p + 1)(p - 3)}\).

Step4: Add the fractions

Add the numerators: \(3p(p - 3)+5(p + 1)\). Expand the numerator: \(3p^2-9p + 5p+5=3p^2-4p + 5\). So the sum is \(\frac{3p^2-4p + 5}{(p + 1)(p - 3)}\) (or expand the denominator: \(\frac{3p^2-4p + 5}{p^2-2p-3}\)).

Answer:

\(\frac{3p^2 - 4p + 5}{(p + 1)(p - 3)}\) (or \(\frac{3p^2 - 4p + 5}{p^2 - 2p - 3}\))