Let's solve the given polynomial expression step-by-step:
We start with:
[tex]\[
(g - h)(g^2 - 3gh + 2h^2)
\][/tex]
To find which expression is equivalent to this, we need to expand and simplify it.
Step 1: Distribute [tex]\( (g - h) \)[/tex] across each term inside the parentheses.
[tex]\[
= g(g^2 - 3gh + 2h^2) - h(g^2 - 3gh + 2h^2)
\][/tex]
Now, distribute [tex]\( g \)[/tex] through the first term and [tex]\( -h \)[/tex] through the second term:
[tex]\[
= g \cdot g^2 + g \cdot (-3gh) + g \cdot 2h^2 - h \cdot g^2 - h \cdot (-3gh) - h \cdot 2h^2
\][/tex]
Step 2: Perform each multiplication.
[tex]\[
= g^3 - 3g^2h + 2gh^2 - hg^2 + 3gh^2 - 2h^3
\][/tex]
Step 3: Combine like terms.
Notice we have two terms containing [tex]\( g^2h \)[/tex], two terms containing [tex]\( gh^2 \)[/tex], and one term of each [tex]\( g^3 \)[/tex] and [tex]\( -2h^3 \)[/tex].
[tex]\[
= g^3 - 3g^2h - g^2h + 2gh^2 + 3gh^2 - 2h^3
\][/tex]
[tex]\[
= g^3 - 4g^2h + 5gh^2 - 2h^3
\][/tex]
Thus, the simplified expanded form of the given polynomial expression is:
[tex]\[
g^3 - 4g^2h + 5gh^2 - 2h^3
\][/tex]
Therefore, the correct choice is:
[tex]\[
\boxed{A: g^3 - 4g^2h + 5gh^2 - 2h^3}
\][/tex]
