Let's simplify the given expression step-by-step:
[tex]\[
\frac{-9a^2 x^2 + 1}{(a - b)(a + b)}
\][/tex]
### Step 1: Simplifying the Numerator
First, look at the numerator:
[tex]\[
-9a^2 x^2 + 1
\][/tex]
Notice that this can be written as:
[tex]\[
-(9a^2 x^2 - 1)
\][/tex]
We recognize that [tex]\( 9a^2 x^2 - 1 \)[/tex] is a difference of squares, because it is of the form [tex]\( a^2 - b^2 \)[/tex] where [tex]\( a = 3ax \)[/tex] and [tex]\( b = 1 \)[/tex]. Recall the difference of squares formula:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Applying this to [tex]\( 9a^2 x^2 - 1 \)[/tex]:
[tex]\[
9a^2 x^2 - 1 = (3ax)^2 - 1^2 = (3ax - 1)(3ax + 1)
\][/tex]
Therefore:
[tex]\[
-(9a^2 x^2 - 1) = -(3ax - 1)(3ax + 1)
\][/tex]
So the numerator becomes:
[tex]\[
-[(3ax - 1)(3ax + 1)]
\][/tex]
### Step 2: Analyzing the Denominator
The denominator is:
[tex]\[
(a - b)(a + b)
\][/tex]
### Step 3: Combining Numerator and Denominator
Now, we can rewrite the original expression with the factored forms of both the numerator and the denominator:
[tex]\[
\frac{-[(3ax - 1)(3ax + 1)]}{(a - b)(a + b)}
\][/tex]
### Step 4: Final Simplification
The expression:
[tex]\[
-[(3ax - 1)(3ax + 1)]
\][/tex]
is simply a representation of the product, so the simplified form of the expression is:
[tex]\[
-\frac{(3ax - 1)(3ax + 1)}{(a - b)(a + b)}
\][/tex]
This is the simplified form of the original given expression:
[tex]\[
-\frac{(3ax - 1)(3ax + 1)}{(a - b)(a + b)}
\][/tex]
