To simplify the expression [tex]\(\frac{a^2 - b^2}{a^2 + ab - a - b}\)[/tex], let's proceed step by step.
### Step 1: Factorize the Numerator
The numerator is [tex]\(a^2 - b^2\)[/tex]. This is a difference of squares, which can be factorized using the formula [tex]\(x^2 - y^2 = (x - y)(x + y)\)[/tex]. Applying this formula:
[tex]\[a^2 - b^2 = (a - b)(a + b)\][/tex]
### Step 2: Factorize the Denominator
The denominator is [tex]\(a^2 + ab - a - b\)[/tex]. To factorize this expression, we can try to rearrange and group terms to make it easier to factor:
[tex]\[a^2 + ab - a - b\][/tex]
Group the terms carefully:
[tex]\[a^2 + ab - a - b = a(a + b) - 1(a + b)\][/tex]
Now factor by grouping:
[tex]\[(a - 1)(a + b)\][/tex]
### Step 3: Simplify the Expression
Now that we have the factorizations, the expression becomes:
[tex]\[\frac{a^2 - b^2}{a^2 + ab - a - b} = \frac{(a - b)(a + b)}{(a - 1)(a + b)}\][/tex]
Notice that [tex]\((a + b)\)[/tex] appears in both the numerator and the denominator, so we can cancel it out, assuming [tex]\(a + b \neq 0\)[/tex]:
[tex]\[\frac{(a - b)(a + b)}{(a - 1)(a + b)} = \frac{a - b}{a - 1}\][/tex]
### Final Simplified Expression
Therefore, the simplified form of the given expression is:
[tex]\[\frac{a - b}{a - 1}\][/tex]
