To simplify the expression [tex]\((2x - 3y)^2 - (2xy + 9y^2)\)[/tex], we will proceed step-by-step.
### Step 1: Expand each term
First, expand the squared term and distribute the other terms properly.
1. Expand [tex]\((2x - 3y)^2\)[/tex]:
[tex]\[
(2x - 3y)^2 = (2x - 3y)(2x - 3y)
\][/tex]
Using the distributive property (FOIL method):
[tex]\[
(2x - 3y)(2x - 3y) = 2x \cdot 2x + 2x \cdot (-3y) + (-3y) \cdot 2x + (-3y) \cdot (-3y)
\][/tex]
[tex]\[
= 4x^2 - 6xy - 6xy + 9y^2
\][/tex]
Combine like terms:
[tex]\[
= 4x^2 - 12xy + 9y^2
\][/tex]
### Step 2: Handle the second term
The term [tex]\(2xy + 9y^2\)[/tex] remains as it is.
### Step 3: Subtract the second term from the first
Now, subtract [tex]\(2xy + 9y^2\)[/tex] from the expanded form of [tex]\((2x - 3y)^2\)[/tex]:
[tex]\[
4x^2 - 12xy + 9y^2 - (2xy + 9y^2)
\][/tex]
Distribute the negative sign inside the parentheses:
[tex]\[
4x^2 - 12xy + 9y^2 - 2xy - 9y^2
\][/tex]
Combine like terms:
[tex]\[
4x^2 - 12xy - 2xy + 9y^2 - 9y^2
\][/tex]
[tex]\[
= 4x^2 - 14xy
\][/tex]
### Step 4: Factor common terms
We notice that we can factor out [tex]\(2x\)[/tex] from the expression:
[tex]\[
4x^2 - 14xy = 2x(2x - 7y)
\][/tex]
### Result
The simplified form of the given expression is:
[tex]\[
2x(2x - 7y)
\][/tex]
