Question

which shows one way to determine the factors of $x^3 + 5x^2 - 6x - 30$ by grouping?
$\bigcirc\\ x(x^2 - 5) + 6(x^2 - 5)$
$\bigcirc\\ x(x^2 + 5) - 6(x^2 + 5)$
$\bigcirc\\ x^2(x - 5) + 6(x - 5)$
$\bigcirc\\ x^2(x + 5) - 6(x + 5)$

Explanation:

Step1: Group the terms

We have the polynomial \(x^{3}+5x^{2}-6x - 30\). Group the first two terms and the last two terms: \((x^{3}+5x^{2})+(-6x - 30)\).

Step2: Factor out the GCF from each group

For the first group \(x^{3}+5x^{2}\), the greatest common factor (GCF) is \(x^{2}\). Factoring out \(x^{2}\), we get \(x^{2}(x + 5)\).

For the second group \(-6x-30\), the GCF is \(- 6\). Factoring out \(-6\), we get \(-6(x + 5)\).

So the polynomial factored by grouping is \(x^{2}(x + 5)-6(x + 5)\).

Answer:

D. \(x^{2}(x + 5)-6(x + 5)\) (assuming the options are labeled A, B, C, D in order as given in the problem)