Question

factor by grouping.
uv - 32u + 4v - 8u²

Explanation:

Step1: Rearrange the terms

First, we rearrange the given expression \( uv - 32u + 4v - 8u^{2} \) to group the terms with common factors. Let's rewrite it as \( uv + 4v - 32u - 8u^{2} \).

Step2: Factor out common factors from each group

Now, we factor out the common factor from the first two terms and the last two terms.
For the first group \( uv + 4v \), the common factor is \( v \). So, \( uv + 4v = v(u + 4) \).
For the second group \( -32u - 8u^{2} \), the common factor is \( -8u \). So, \( -32u - 8u^{2} = -8u(4 + u) \). Notice that \( 4 + u \) is the same as \( u + 4 \).

Step3: Factor out the common binomial factor

Now, we have the expression \( v(u + 4) - 8u(u + 4) \). Here, the common binomial factor is \( (u + 4) \). We factor that out:
\( (u + 4)(v - 8u) \)

Answer:

\( (u + 4)(v - 8u) \)