Certainly! Let's take a detailed, step-by-step approach to solve these two expressions and verify their expansion.
### Expression 1: [tex]\((2x - 3y)(4x^2 + 6xy + 9y^2)\)[/tex]
We'll start by distributing [tex]\((2x - 3y)\)[/tex] across each term in [tex]\((4x^2 + 6xy + 9y^2)\)[/tex]:
1. First term: [tex]\(2x \cdot (4x^2 + 6xy + 9y^2)\)[/tex]
[tex]\[
2x \cdot 4x^2 = 8x^3
\][/tex]
[tex]\[
2x \cdot 6xy = 12x^2y
\][/tex]
[tex]\[
2x \cdot 9y^2 = 18xy^2
\][/tex]
Adding these, we get [tex]\(8x^3 + 12x^2y + 18xy^2\)[/tex].
2. Second term: [tex]\(-3y \cdot (4x^2 + 6xy + 9y^2)\)[/tex]
[tex]\[
-3y \cdot 4x^2 = -12x^2y
\][/tex]
[tex]\[
-3y \cdot 6xy = -18xy^2
\][/tex]
[tex]\[
-3y \cdot 9y^2 = -27y^3
\][/tex]
Adding these, we get [tex]\(-12x^2y - 18xy^2 - 27y^3\)[/tex].
3. Combining all terms:
[tex]\[
(8x^3 + 12x^2y + 18xy^2) + (-12x^2y - 18xy^2 - 27y^3)
\][/tex]
Simplify by combining like terms:
[tex]\[
8x^3 + (12x^2y - 12x^2y) + (18xy^2 - 18xy^2) - 27y^3 = 8x^3 - 27y^3
\][/tex]
### Expression 2: [tex]\(4xy(2x - 3y)(2x + 3y)\)[/tex]
1. First, consider [tex]\((2x - 3y)(2x + 3y)\)[/tex]:
Notice that this expression is in the form of a difference of squares: [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
Here, [tex]\(a = 2x\)[/tex] and [tex]\(b = 3y\)[/tex]:
[tex]\[
(2x - 3y)(2x + 3y) = (2x)^2 - (3y)^2 = 4x^2 - 9y^2
\][/tex]
2. Next, distribute [tex]\(4xy\)[/tex] across the result [tex]\(4x^2 - 9y^2\)[/tex]:
[tex]\[
4xy \cdot 4x^2 = 16x^3y
\][/tex]
[tex]\[
4xy \cdot (-9y^2) = -36xy^3
\][/tex]
So, combining these, we get:
[tex]\[
16x^3y - 36xy^3
\][/tex]
### Final Results:
1. For [tex]\((2x - 3y)(4x^2 + 6xy + 9y^2)\)[/tex]:
[tex]\[
8x^3 - 27y^3
\][/tex]
2. For [tex]\(4xy(2x - 3y)(2x + 3y)\)[/tex]:
[tex]\[
16x^3y - 36xy^3
\][/tex]
Thus, the expanded forms of the given expressions are:
[tex]\[
(2x - 3y)(4x^2 + 6xy + 9y^2) = 8x^3 - 27y^3
\][/tex]
[tex]\[
4xy(2x - 3y)(2x + 3y) = 16x^3y - 36xy^3
\][/tex]
