To determine which expression is equivalent to the given polynomial expression, [tex]\((g-h)(g^2 - 3gh + 2h^2)\)[/tex], we need to expand and simplify it step by step.
Let's begin by performing the multiplication:
1. First Distribution:
[tex]\[
(g - h) \cdot (g^2 - 3gh + 2h^2)
\][/tex]
2. Distribute [tex]\(g\)[/tex]:
[tex]\[
g \cdot (g^2 - 3gh + 2h^2) = g \cdot g^2 + g \cdot (-3gh) + g \cdot 2h^2 = g^3 - 3g^2h + 2gh^2
\][/tex]
3. Distribute [tex]\(-h\)[/tex]:
[tex]\[
-h \cdot (g^2 - 3gh + 2h^2) = -h \cdot g^2 + (-h) \cdot (-3gh) + (-h) \cdot 2h^2 = -hg^2 + 3gh^2 - 2h^3
\][/tex]
4. Combine Like Terms:
[tex]\[
g^3 - 3g^2h + 2gh^2 - hg^2 + 3gh^2 - 2h^3
\][/tex]
5. Simplify:
- Combine the [tex]\(g^2h\)[/tex] terms:
[tex]\[
- 3g^2h - hg^2 = -4g^2h
\][/tex]
- Combine the [tex]\(gh^2\)[/tex] terms:
[tex]\[
2gh^2 + 3gh^2 = 5gh^2
\][/tex]
6. Final Expression:
[tex]\[
g^3 - 4g^2h + 5gh^2 - 2h^3
\][/tex]
Therefore, the expression that matches our simplified polynomial is:
C. [tex]\(g^3 - 4g^2h + 5gh^2 - 2h^3\)[/tex]
