Let's start by factoring out the greatest common factor. Then, we will factor the resulting trinomial.
To factor a quadratic expression with a leading coefficient of $1,$ we first need to identify the values of $b$ and $c.$ $\begin{aligned} \textbf{General Expression:}&\ m^2+\textcolor{#ff00ff}{b}m+\textcolor{#ff8c00}{c} \\ \textbf{Our Expression:}&\ m^2\textcolor{#ff00ff}{-12}m +\textcolor{#ff8c00}{32} \end{aligned}$ Next, we have to find a factor pair of $\textcolor{#ff8c00}{c}$ $=$ $\textcolor{#ff8c00}{32}$ whose sum is $\textcolor{#ff00ff}{b}$ $=$ $\textcolor{#ff00ff}{\text{-} 12}.$ Note that $32$ is a positive number, so for the product of the factors to be positive, they must have the same sign — both positive or both negative.
Factor Pair | Product of Factors | Sum of Factors |
---|---|---|
$1$ and $32$ | $\textcolor{#ff8c00}{32}$ | $33$ |
$\text{-}1$ and $\text{-}32$ | $\textcolor{#ff8c00}{32}$ | $\text{-} 33$ |
$2$ and $16$ | $\textcolor{#ff8c00}{32}$ | $18$ |
$\text{-}2$ and $\text{-}16$ | $\textcolor{#ff8c00}{32}$ | $\text{-} 18$ |
$4$ and $8$ | $\textcolor{#ff8c00}{32}$ | $12$ |
$\text{-}4$ and $\text{-}8$ | $\textcolor{#ff8c00}{32}$ | $\textcolor{#ff00ff}{\text{-}12}$ |
The factors whose product is $\textcolor{#ff8c00}{32}$ and whose sum is $\textcolor{#ff00ff}{\text{-}12}$ are ${\color{#0000FF}{\text{-} 4}}$ and $\textcolor{deepskyblue}{\text{-} 8}.$ With this information, we can now factor the trinomial. $\begin{gathered} m^2\textcolor{#ff00ff}{-12}m+\textcolor{#ff8c00}{32}\quad\Leftrightarrow\quad (m{\color{#0000FF}{\,-\,4}})(m-\textcolor{deepskyblue}{8}) \end{gathered}$ Before we finish, remember that we factored out the greatest common factor from the original expression. Therefore, we need to include it again. $\begin{gathered} 2m^6-24m^5+64m^4 \quad \Leftrightarrow \quad 2m^4(m-4)(m-8) \end{gathered}$