Question

which shows one way to determine the factors of $x^3 + 4x^2 + 5x + 20$ by grouping?
$\bigcirc\\ x(x^2 + 4) + 5(x^2 + 4)$
$\bigcirc\\ x^2(x + 4) + 5(x + 4)$
$\bigcirc\\ x^2(x + 5) + 4(x + 5)$
$\bigcirc\\ x(x^2 + 5) + 4x(x^2 + 5)$

Explanation:

Step1: Recall factoring by grouping

To factor by grouping, we group terms in the polynomial and factor out the greatest common factor (GCF) from each group. The polynomial is \(x^{3}+4x^{2}+5x + 20\). We can group the first two terms and the last two terms: \((x^{3}+4x^{2})+(5x + 20)\).

Step2: Factor out GCF from each group

For the first group \(x^{3}+4x^{2}\), the GCF is \(x^{2}\). Factoring out \(x^{2}\) gives \(x^{2}(x + 4)\). For the second group \(5x+20\), the GCF is \(5\). Factoring out \(5\) gives \(5(x + 4)\). So the factoring by grouping gives \(x^{2}(x + 4)+5(x + 4)\).

Answer:

\(x^{2}(x + 4)+5(x + 4)\) (the second option: \(x^{2}(x + 4)+5(x + 4)\))