Sure, let's solve the expression [tex]\(\left(\frac{a^2 - 2a - 3}{a + 1}\right)\)[/tex] step by step.
### Step 1: Factor the Numerator
First, we need to factor the numerator [tex]\(a^2 - 2a - 3\)[/tex].
To factor [tex]\(a^2 - 2a - 3\)[/tex], we look for two numbers that multiply to [tex]\(-3\)[/tex] (the constant term) and add to [tex]\(-2\)[/tex] (the coefficient of [tex]\(a\)[/tex]):
- The pair of numbers that satisfy these conditions are [tex]\(-3\)[/tex] and [tex]\(1\)[/tex]:
[tex]\[a^2 - 2a - 3 = (a - 3)(a + 1)\][/tex]
### Step 2: Rewrite the Fraction
Now that we have factored the numerator, we can rewrite the fraction as:
[tex]\[
\left(\frac{(a - 3)(a + 1)}{a + 1}\right)
\][/tex]
### Step 3: Simplify the Fraction
Next, we notice that [tex]\((a + 1)\)[/tex] in the numerator and denominator can be canceled out, provided [tex]\(a \neq -1\)[/tex] (since division by zero is undefined). Thus, we have:
[tex]\[
\left(\frac{(a - 3)\cancel{(a + 1)}}{\cancel{a + 1}}\right) = a - 3
\][/tex]
### Conclusion
After canceling the common factor [tex]\((a + 1)\)[/tex], we are left with:
[tex]\[
a - 3
\][/tex]
So, the simplified form of the given expression [tex]\(\left(\frac{a^2 - 2a - 3}{a + 1}\right)\)[/tex] is:
[tex]\[
a - 3
\][/tex]
