Sure, let's solve the given expression step-by-step.
We want to expand and simplify the expression:
[tex]\[
(2xy + 3z) \left( 4x^2y^2 - 6xyz + 9z^2 \right)
\][/tex]
To do this, we will use the distributive property (also known as the FOIL method for binomials):
1. Multiply each term in the first factor [tex]\((2xy + 3z)\)[/tex] by each term in the second factor [tex]\((4x^2y^2 - 6xyz + 9z^2)\)[/tex].
Let's distribute [tex]\(2xy\)[/tex] through the second factor:
[tex]\[
2xy \cdot (4x^2y^2) = 2xy \cdot 4x^2y^2 = 8x^3y^3
\][/tex]
[tex]\[
2xy \cdot (-6xyz) = 2xy \cdot (-6xyz) = -12x^2y^2z
\][/tex]
[tex]\[
2xy \cdot 9z^2 = 2xy \cdot 9z^2 = 18xyz^2
\][/tex]
Next, let's distribute [tex]\(3z\)[/tex] through the second factor:
[tex]\[
3z \cdot (4x^2y^2) = 3z \cdot 4x^2y^2 = 12x^2y^2z
\][/tex]
[tex]\[
3z \cdot (-6xyz) = 3z \cdot (-6xyz) = -18xyz^2
\][/tex]
[tex]\[
3z \cdot 9z^2 = 3z \cdot 9z^2 = 27z^3
\][/tex]
Now, let's combine all these terms together:
[tex]\[
8x^3y^3 + 18xyz^2 - 12x^2y^2z + 12x^2y^2z - 18xyz^2 + 27z^3
\][/tex]
Next, we can combine like terms. Notice that [tex]\(+18xyz^2\)[/tex] and [tex]\(-18xyz^2\)[/tex] cancel each other out, and [tex]\(+12x^2y^2z\)[/tex] and [tex]\(-12x^2y^2z\)[/tex] cancel each other out as well:
[tex]\[
8x^3y^3 + 27z^3
\][/tex]
So, the expanded and simplified form of [tex]\((2xy + 3z) \left( 4x^2y^2 - 6xyz + 9z^2 \right)\)[/tex] is:
[tex]\[
8x^3y^3 + 27z^3
\][/tex]
This is the final answer.
