Question

reduce the rational expression to lowest terms.\\(\frac{4s^{2}+28s + 48}{16s^{2}-32s - 240}\\)\\(\frac{4s^{2}+28s + 48}{16s^{2}-32s - 240}=\square\\)

Explanation:

Step1: Factor numerator and denominator

First, factor the numerator \(4s^2 + 28s + 48\). We can factor out a 4 first: \(4(s^2 + 7s + 12)\). Then factor the quadratic: \(4(s + 3)(s + 4)\).

Next, factor the denominator \(16s^2 - 32s - 240\). Factor out a 16: \(16(s^2 - 2s - 15)\). Then factor the quadratic: \(16(s - 5)(s + 3)\).

So now the expression is \(\frac{4(s + 3)(s + 4)}{16(s - 5)(s + 3)}\).

Step2: Cancel common factors

We can cancel the common factors of 4 and \((s + 3)\) (assuming \(s
eq -3\) to avoid division by zero).

Dividing 4 and 16 by 4 gives \(\frac{1}{4}\), and canceling \((s + 3)\) leaves us with \(\frac{s + 4}{4(s - 5)}\) or we can write it as \(\frac{s + 4}{4s - 20}\) after distributing the 4 in the denominator.

Answer:

\(\frac{s + 4}{4(s - 5)}\) (or \(\frac{s + 4}{4s - 20}\))