Let's simplify the given polynomial expression step-by-step:
Given expression:
[tex]\[
(g - h) \left(g^2 - 3gh + 2h^2\right)
\][/tex]
1. Distribute [tex]\((g - h)\)[/tex] across [tex]\(\left(g^2 - 3gh + 2h^2\right)\)[/tex]:
[tex]\[
(g - h)(g^2 - 3gh + 2h^2) = g(g^2 - 3gh + 2h^2) - h(g^2 - 3gh + 2h^2)
\][/tex]
2. Distribute [tex]\(g\)[/tex] in the first term:
[tex]\[
g(g^2 - 3gh + 2h^2) = g \cdot g^2 + g \cdot (-3gh) + g \cdot 2h^2 = g^3 - 3g^2h + 2gh^2
\][/tex]
3. Distribute [tex]\(-h\)[/tex] in the second term:
[tex]\[
-h(g^2 - 3gh + 2h^2) = -h \cdot g^2 + h \cdot (3gh) - h \cdot 2h^2 = -hg^2 + 3gh^2 - 2h^3
\][/tex]
4. Combine the results from steps 2 and 3:
[tex]\[
g^3 - 3g^2h + 2gh^2 - hg^2 + 3gh^2 - 2h^3
\][/tex]
5. Simplify by combining like terms:
[tex]\[
g^3 - 3g^2h - hg^2 + 2gh^2 + 3gh^2 - 2h^3 = g^3 - 4g^2h + 5gh^2 - 2h^3
\][/tex]
Thus, the expression equivalent to [tex]\((g - h)(g^2 - 3gh + 2h^2)\)[/tex] is:
[tex]\[
g^3 - 4g^2h + 5gh^2 - 2h^3
\][/tex]
Therefore, the correct answer is:
A. [tex]\(g^3 - 4g^2h + 5gh^2 - 2h^3\)[/tex]
