Of course! Let's multiply the expressions [tex]\((x^2 - 2xy)\)[/tex] and [tex]\((3xy + y^2)\)[/tex] step by step.
### Step-by-Step Multiplication
1. Distribute [tex]\( x^2 \)[/tex] across [tex]\(3xy + y^2\)[/tex]:
[tex]\[
x^2 \cdot (3xy + y^2) = x^2 \cdot 3xy + x^2 \cdot y^2
\][/tex]
This gives:
[tex]\[
3x^3y + x^2y^2
\][/tex]
2. Distribute [tex]\(-2xy\)[/tex] across [tex]\(3xy + y^2\)[/tex]:
[tex]\[
-2xy \cdot (3xy + y^2) = -2xy \cdot 3xy - 2xy \cdot y^2
\][/tex]
This gives:
[tex]\[
-6x^2y^2 - 2xy^3
\][/tex]
3. Combine all the terms together:
[tex]\[
3x^3y + x^2y^2 - 6x^2y^2 - 2xy^3
\][/tex]
4. Simplify by combining like terms:
[tex]\[
3x^3y + (x^2y^2 - 6x^2y^2) - 2xy^3
\][/tex]
[tex]\[
3x^3y - 5x^2y^2 - 2xy^3
\][/tex]
### Final Result
Therefore, the result of multiplying [tex]\((x^2 - 2xy)\)[/tex] by [tex]\((3xy + y^2)\)[/tex] is:
[tex]\[
3x^3y - 5x^2y^2 - 2xy^3
\][/tex]
