Alright, let’s simplify the given expression step-by-step:
[tex]\[
\frac{a^2 + b^2 + 2ab}{a^2 - b^2}
\][/tex]
1. Identify Useful Identities:
- Recognize that the numerator [tex]\(a^2 + b^2 + 2ab\)[/tex] can be factored using the square of a binomial identity:
[tex]\[
a^2 + b^2 + 2ab = (a + b)^2
\][/tex]
- Recognize that the denominator [tex]\(a^2 - b^2\)[/tex] can be factored using the difference of squares identity:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
2. Substitute the Factored Forms:
- Replace the numerator and the denominator in the expression with their factored forms:
[tex]\[
\frac{(a + b)^2}{(a - b)(a + b)}
\][/tex]
3. Simplify the Expression:
- Notice that [tex]\((a + b)\)[/tex] is a common factor in both the numerator and the denominator. We can cancel out this common factor:
[tex]\[
\frac{(a + b)^2}{(a - b)(a + b)} = \frac{(a + b) \cdot (a + b)}{(a - b) \cdot (a + b)} = \frac{a + b}{a - b}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{a + b}{a - b}
\][/tex]
