Sure! Let's simplify the given expression step-by-step.
We need to simplify:
[tex]\[
\frac{a^2+6a+8}{a^2-16}
\][/tex]
1. First, factor the numerator and the denominator, if possible.
2. Factor the numerator:
The numerator is [tex]\( a^2 + 6a + 8 \)[/tex].
To factor it, we look for two numbers that multiply to 8 (the constant term) and add to 6 (the coefficient of the linear term).
The numbers 2 and 4 fit this requirement because [tex]\( 2 \times 4 = 8 \)[/tex] and [tex]\( 2 + 4 = 6 \)[/tex].
Thus, we can write:
[tex]\[
a^2 + 6a + 8 = (a + 2)(a + 4)
\][/tex]
3. Factor the denominator:
The denominator is [tex]\( a^2 - 16 \)[/tex].
Notice that this is a difference of squares, which has the form [tex]\( a^2 - b^2 = (a + b)(a - b) \)[/tex].
In our case, [tex]\( a^2 - 16 \)[/tex] can be written as [tex]\( (a)^2 - (4)^2 \)[/tex].
Thus, we can factor it as:
[tex]\[
a^2 - 16 = (a + 4)(a - 4)
\][/tex]
4. Substitute the factored forms back into the expression:
[tex]\[
\frac{a^2 + 6a + 8}{a^2 - 16} = \frac{(a + 2)(a + 4)}{(a + 4)(a - 4)}
\][/tex]
5. Simplify the expression:
Notice that [tex]\( (a + 4) \)[/tex] is a common factor in both the numerator and the denominator, so we can cancel it out:
[tex]\[
\frac{(a + 2)(a + 4)}{(a + 4)(a - 4)} = \frac{a + 2}{a - 4} \quad \text{(for \( a \neq -4 \) and \( a \neq 4 \))}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{a + 2}{a - 4}
\][/tex]
Note: The restrictions [tex]\( a \neq 4 \)[/tex] and [tex]\( a \neq -4 \)[/tex] come from the denominators in the original and simplified forms of the expression.
