Sure, let's go through the simplification of the given expression step-by-step:
We start with the expression:
[tex]\[
\frac{x^2 - b^2}{a + b^2}
\][/tex]
### Step 1: Recognize the Difference of Squares
The numerator [tex]\(x^2 - b^2\)[/tex] is a difference of squares. Recall the difference of squares formula:
[tex]\[
x^2 - b^2 = (x + b)(x - b)
\][/tex]
Thus, we can rewrite the numerator as:
[tex]\[
(x + b)(x - b)
\][/tex]
### Step 2: Rewrite the Initial Expression
Substitute the factored form of the numerator back into the original expression:
[tex]\[
\frac{(x + b)(x - b)}{a + b^2}
\][/tex]
### Step 3: Simplify the Expression
Currently, there are no common factors in the numerator and denominator that can be cancelled since the denominator [tex]\(a + b^2\)[/tex] does not share any common terms with the factored form [tex]\((x + b)(x - b)\)[/tex].
Therefore, the simplified form of the expression is:
[tex]\[
\frac{(x + b)(x - b)}{a + b^2}
\][/tex]
### Step 4: Confirm the Simplified Form
The final simplified form of the expression, step-by-step, is:
[tex]\[
\frac{x^2 - b^2}{a + b^2}
\][/tex]
### Step 5: Present the Result
So, the simplified form of the given expression is:
[tex]\[
\frac{-b^2 + x^2}{a + b^2}
\][/tex]
Notice that [tex]\(-b^2 + x^2\)[/tex] is algebraically equivalent to [tex]\(x^2 - b^2\)[/tex], which matches our original term.
Thus, the final answer is:
[tex]\[
\frac{-b^2 + x^2}{a + b^2}
\][/tex]
This completes our step-by-step solution.
