Sure, let's solve the expression [tex]\((3x + 2y)(5x^2 + 2xy - 4y^2)\)[/tex] step-by-step.
We will use the distributive property to expand this expression.
1. First, distribute each term in the first parenthesis [tex]\((3x + 2y)\)[/tex] to each term in the second parenthesis [tex]\((5x^2 + 2xy - 4y^2)\)[/tex].
[tex]\[
(3x + 2y)(5x^2 + 2xy - 4y^2)
\][/tex]
This can be broken down into:
[tex]\[
3x(5x^2 + 2xy - 4y^2) + 2y(5x^2 + 2xy - 4y^2)
\][/tex]
2. Now, distribute [tex]\(3x\)[/tex] to each term inside the parentheses first:
[tex]\[
3x \cdot 5x^2 = 15x^3
\][/tex]
[tex]\[
3x \cdot 2xy = 6x^2y
\][/tex]
[tex]\[
3x \cdot (-4y^2) = -12xy^2
\][/tex]
So, the first part simplifies to:
[tex]\[
15x^3 + 6x^2y - 12xy^2
\][/tex]
3. Next, distribute [tex]\(2y\)[/tex] to each term inside the parentheses:
[tex]\[
2y \cdot 5x^2 = 10x^2y
\][/tex]
[tex]\[
2y \cdot 2xy = 4xy^2
\][/tex]
[tex]\[
2y \cdot (-4y^2) = -8y^3
\][/tex]
So, the second part simplifies to:
[tex]\[
10x^2y + 4xy^2 - 8y^3
\][/tex]
4. Combine the results from both parts:
[tex]\[
15x^3 + 6x^2y - 12xy^2 + 10x^2y + 4xy^2 - 8y^3
\][/tex]
5. Combine like terms:
Combine terms involving [tex]\(x^2y\)[/tex]:
[tex]\[
6x^2y + 10x^2y = 16x^2y
\][/tex]
Combine terms involving [tex]\(xy^2\)[/tex]:
[tex]\[
-12xy^2 + 4xy^2 = -8xy^2
\][/tex]
So, the expression simplifies to:
[tex]\[
15x^3 + 16x^2y - 8xy^2 - 8y^3
\][/tex]
Therefore, the expanded form of the expression [tex]\((3x + 2y)(5x^2 + 2xy - 4y^2)\)[/tex] is:
[tex]\[
15x^3 + 16x^2y - 8xy^2 - 8y^3
\][/tex]
