Sure, let's solve the given expression step-by-step.
First, let's rewrite the expressions for clarity:
Expression 1:
[tex]\[ 4x^3 - 2x^2 y + 6xy^2 + 6xy^2 \][/tex]
Expression 2:
[tex]\[ x^2 y - x^2 y - xy^2 - 2y^3 \][/tex]
### Simplify Expression 1:
Combine like terms in the first expression:
[tex]\[ 4x^3 - 2x^2 y + 6xy^2 + 6xy^2 \][/tex]
[tex]\[ = 4x^3 - 2x^2 y + 12xy^2 \][/tex]
This is the simplified form of Expression 1.
### Simplify Expression 2:
Combine like terms in the second expression:
[tex]\[ x^2 y - x^2 y - xy^2 - 2y^3 \][/tex]
[tex]\[ = 0 - xy^2 - 2y^3 \][/tex]
[tex]\[ = -xy^2 - 2y^3 \][/tex]
This is the simplified form of Expression 2.
Now, we need to multiply these simplified expressions:
[tex]\[ (4x^3 - 2x^2 y + 12xy^2)(-xy^2 - 2y^3) \][/tex]
### Multiply the Expressions Term-by-Term:
We will distribute each term in the first expression to each term in the second expression.
#### Step 1: Multiply [tex]\(4x^3\)[/tex] by each term in [tex]\(-xy^2 - 2y^3\)[/tex]:
[tex]\[ 4x^3 \cdot (-xy^2) = -4x^4 y^2 \][/tex]
[tex]\[ 4x^3 \cdot (-2y^3) = -8x^3 y^3 \][/tex]
#### Step 2: Multiply [tex]\(-2x^2 y\)[/tex] by each term in [tex]\(-xy^2 - 2y^3\)[/tex]:
[tex]\[ (-2x^2 y) \cdot (-xy^2) = 2x^3 y^3 \][/tex]
[tex]\[ (-2x^2 y) \cdot (-2y^3) = 4x^2 y^4 \][/tex]
#### Step 3: Multiply [tex]\(12xy^2\)[/tex] by each term in [tex]\(-xy^2 - 2y^3\)[/tex]:
[tex]\[ 12xy^2 \cdot (-xy^2) = -12x^2 y^4 \][/tex]
[tex]\[ 12xy^2 \cdot (-2y^3) = -24xy^5 \][/tex]
### Combine All the Terms:
Now, add all the products we found:
[tex]\[ -4x^4 y^2 - 8x^3 y^3 + 2x^3 y^3 + 4x^2 y^4 - 12x^2 y^4 - 24xy^5 \][/tex]
Combine like terms where possible:
[tex]\[ -4x^4 y^2 + (-8x^3 y^3 + 2x^3 y^3) + (4x^2 y^4 - 12x^2 y^4) - 24xy^5 \][/tex]
[tex]\[ = -4x^4 y^2 - 6x^3 y^3 - 8x^2 y^4 - 24xy^5 \][/tex]
So the final result of the given expression is:
[tex]\[ \boxed{-4x^4 y^2 - 6x^3 y^3 - 8x^2 y^4 - 24xy^5} \][/tex]
