Question

factor completely.
$2w^2 + 18w + 30$

Explanation:

Step1: Factor out the GCF

First, find the greatest common factor (GCF) of the terms \(2w^2\), \(18w\), and \(30\). The GCF of 2, 18, and 30 is 2. So we factor out 2:
\(2w^2 + 18w + 30 = 2(w^2 + 9w + 15)\)
Wait, actually, let's check the discriminant of the quadratic inside the parentheses. The quadratic is \(w^2 + 9w + 15\). The discriminant \(D = 9^2 - 4\times1\times15 = 81 - 60 = 21\), which is not a perfect square. Wait, maybe I made a mistake in the GCF? Wait, 2, 18, 30: 2 divides 2, 18, 30. But let's check the original polynomial again. Wait, maybe the problem is \(2w^2 + 18w + 30\). Wait, let's factor 2 first: \(2(w^2 + 9w + 15)\). But maybe there's a mistake in the problem, or maybe I miscalculated. Wait, no, maybe the quadratic can be factored further? Wait, no, since the discriminant is 21, which is not a perfect square, so the quadratic \(w^2 + 9w + 15\) doesn't factor over the integers. Wait, but maybe the original problem was supposed to be \(2w^2 + 18w + 40\)? Let's check: 2, 18, 40. GCF is 2. Then \(w^2 + 9w + 20\), which factors into \((w + 4)(w + 5)\). But the given problem is 30. Wait, maybe I made a mistake. Wait, let's re-express the original problem: \(2w^2 + 18w + 30\). Let's factor 2: \(2(w^2 + 9w + 15)\). Since 15 factors into 1×15 or 3×5. 1 + 15 = 16, 3 + 5 = 8, neither is 9. So the quadratic \(w^2 + 9w + 15\) is irreducible over the integers. So the complete factorization is \(2(w^2 + 9w + 15)\). Wait, but maybe the problem has a typo. Alternatively, maybe I misread the coefficients. Let me check again: \(2w^2 + 18w + 30\). Yes, that's what's given. So the GCF is 2, and the quadratic inside doesn't factor further over the integers. So the complete factorization is \(2(w^2 + 9w + 15)\).

Wait, but maybe the problem is correct, and we can factor it as 2 times the quadratic, which is as far as we can go. So the final factored form is \(2(w^2 + 9w + 15)\). But let me confirm. Let's expand \(2(w^2 + 9w + 15)\): \(2w^2 + 18w + 30\), which matches the original. So that's the complete factorization.

Answer:

\(2(w^2 + 9w + 15)\)