Certainly! Let's solve the given mathematical problem step by step.
We are given the rational expression:
[tex]\[ \frac{11(x + 8)}{(x + 1)(x + 8)} \][/tex]
and need to simplify it under the condition that [tex]\( x \neq -1 \)[/tex] or [tex]\( x \neq -8 \)[/tex].
Here’s the step-by-step breakdown of how to simplify the expression:
1. Identify common factors:
The numerator of the expression is [tex]\( 11(x + 8) \)[/tex].
The denominator of the expression is [tex]\( (x + 1)(x + 8) \)[/tex].
We need to look for common factors in both the numerator and the denominator.
2. Cancel out common factors:
Notice that both the numerator [tex]\( 11(x + 8) \)[/tex] and the denominator [tex]\( (x + 8)(x + 1) \)[/tex] have the factor [tex]\( (x + 8) \)[/tex] in common.
Therefore, we can cancel out [tex]\( x + 8 \)[/tex] from both the numerator and the denominator:
[tex]\[
\frac{11(x + 8)}{(x + 1)(x + 8)} = \frac{11 \cancel{(x + 8)}}{(x + 1) \cancel{(x + 8)}} = \frac{11}{x + 1}
\][/tex]
3. Simplified expression:
After canceling the common factor [tex]\( x + 8 \)[/tex], we are left with:
[tex]\[
\frac{11}{x + 1}
\][/tex]
4. Conclusion:
From the simplified expression, we can see that the rational expression [tex]\( \frac{11(x + 8)}{(x + 1)(x + 8)} \)[/tex] simplifies to [tex]\( \frac{11}{x + 1} \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{\text{D. } \frac{11}{x + 1}} \][/tex]
