Question

factor completely.\
$-2w^2 - w + 21$

Explanation:

Step1: Factor out -1

First, we factor out -1 from the entire quadratic expression to make the leading coefficient positive.
$$-2w^2 - w + 21 = - (2w^2 + w - 21)$$

Step2: Factor the quadratic inside the parentheses

We need to factor \(2w^2 + w - 21\). We look for two numbers that multiply to \(2\times(-21)= - 42\) and add up to 1. The numbers are 7 and -6.
We rewrite the middle term using these numbers:
$$2w^2 + 7w - 6w - 21$$
Then we group the terms:
$$(2w^2 + 7w) + (-6w - 21)$$
Factor out the greatest common factor from each group:
$$w(2w + 7) - 3(2w + 7)$$
Now we can factor out \((2w + 7)\):
$$(2w + 7)(w - 3)$$

Step3: Combine with the -1 factored out earlier

Since we had factored out -1 in Step 1, we now have:
$$- (2w + 7)(w - 3)=(-2w - 7)(w - 3)$$ or we can also write it as \((2w + 7)(-w + 3)\)

Answer:

\(-(2w + 7)(w - 3)\) (or equivalent forms like \((-2w - 7)(w - 3)\) or \((2w + 7)(-w + 3)\))