To simplify the expression
[tex]\[
\frac{6}{b+1} + \frac{3}{b+9}
\][/tex]
we need to combine these two rational expressions into one by finding a common denominator.
1. Identify the common denominator: The common denominator of [tex]\(b+1\)[/tex] and [tex]\(b+9\)[/tex] is [tex]\((b+1)(b+9)\)[/tex].
2. Rewrite each fraction with the common denominator:
[tex]\[
\frac{6}{b+1} = \frac{6(b+9)}{(b+1)(b+9)}
\][/tex]
[tex]\[
\frac{3}{b+9} = \frac{3(b+1)}{(b+1)(b+9)}
\][/tex]
3. Combine the fractions:
[tex]\[
\frac{6(b+9)}{(b+1)(b+9)} + \frac{3(b+1)}{(b+1)(b+9)}
\][/tex]
4. Simplify the numerator:
[tex]\[
\frac{6(b+9) + 3(b+1)}{(b+1)(b+9)}
\][/tex]
Expand the terms in the numerator:
[tex]\[
6(b+9) = 6b + 54
\][/tex]
[tex]\[
3(b+1) = 3b + 3
\][/tex]
Add these together:
[tex]\[
6b + 54 + 3b + 3 = 9b + 57
\][/tex]
5. Combine the numerator over the common denominator:
[tex]\[
\frac{9b + 57}{(b+1)(b+9)}
\][/tex]
6. Factor out the greatest common factor in the numerator, which is 3:
[tex]\[
9b + 57 = 3(3b + 19)
\][/tex]
7. Write the final simplified expression:
[tex]\[
\frac{3(3b + 19)}{(b+1)(b+9)}
\][/tex]
So, the completely factored and simplified form of the expression
[tex]\[
\frac{6}{b+1} + \frac{3}{b+9}
\][/tex]
is
[tex]\[
\frac{3(3b + 19)}{(b+1)(b+9)}
\][/tex]
