Sure! Let's solve the expression step by step. We need to expand the following expression:
[tex]\[
(2xy + 3z)(4x^2y^2 - 6xyz + 9z^2)
\][/tex]
### Step 1: Distribute [tex]\(2xy\)[/tex] across the second term
[tex]\[
2xy \cdot (4x^2y^2 - 6xyz + 9z^2)
\][/tex]
Expanding this, we get:
[tex]\[
2xy \cdot 4x^2y^2 - 2xy \cdot 6xyz + 2xy \cdot 9z^2
\][/tex]
Simplifying each term individually:
[tex]\[
= 8x^3y^3 - 12x^2y^2z + 18xyz^2
\][/tex]
### Step 2: Distribute [tex]\(3z\)[/tex] across the second term
[tex]\[
3z \cdot (4x^2y^2 - 6xyz + 9z^2)
\][/tex]
Expanding this, we get:
[tex]\[
3z \cdot 4x^2y^2 - 3z \cdot 6xyz + 3z \cdot 9z^2
\][/tex]
Simplifying each term individually:
[tex]\[
= 12x^2y^2z - 18xyz^2 + 27z^3
\][/tex]
### Step 3: Combine all the expanded terms and simplify if necessary
Combining all the terms from Step 1 and Step 2:
[tex]\[
8x^3y^3 - 12x^2y^2z + 18xyz^2 + 12x^2y^2z - 18xyz^2 + 27z^3
\][/tex]
Now, let's combine like terms:
- [tex]\( -12x^2y^2z \)[/tex] and [tex]\( +12x^2y^2z \)[/tex] cancel each other out.
- [tex]\( +18xyz^2 \)[/tex] and [tex]\( -18xyz^2 \)[/tex] cancel each other out.
So, we are left with:
[tex]\[
8x^3y^3 + 27z^3
\][/tex]
Thus, the expanded form of the given expression is:
[tex]\[
8x^3y^3 + 27z^3
\][/tex]
