Question

factor by grouping: ( x^3 + 7x^2 - 2x - 14 )
( (x^2 - 2)(x + 7) )
( (x^2 + 2)(x + 7) )
( (x^2 - 2)(x - 7) )
( (x^2 + 2)(x - 7) )

Explanation:

Step1: Group the terms

We have the polynomial \(x^{3}+7x^{2}-2x - 14\). Group the first two terms and the last two terms: \((x^{3}+7x^{2})+(-2x - 14)\)

Step2: Factor out the GCF from each group

For the first group \(x^{3}+7x^{2}\), the greatest common factor (GCF) is \(x^{2}\), so we factor it out: \(x^{2}(x + 7)\)
For the second group \(-2x-14\), the GCF is \(-2\), so we factor it out: \(-2(x + 7)\)
Now the polynomial becomes \(x^{2}(x + 7)-2(x + 7)\)

Step3: Factor out the common binomial factor

We can see that \((x + 7)\) is a common factor, so we factor it out: \((x^{2}-2)(x + 7)\)

Answer:

\((x^{2}-2)(x + 7)\) (the first option in the given choices)