To simplify the expression [tex]\(\frac{b-5}{2b} \cdot \frac{b^2 + 3b}{b-5}\)[/tex], let's follow these steps:
1. Write down the given expression:
[tex]\[
\frac{b-5}{2b} \cdot \frac{b^2 + 3b}{b-5}
\][/tex]
2. Notice that [tex]\((b - 5)\)[/tex] appears in both the numerator and the denominator, so they cancel each other out (as long as [tex]\(b \neq 5\)[/tex]).
After canceling [tex]\((b - 5)\)[/tex]:
[tex]\[
\frac{1}{2b} \cdot (b^2 + 3b)
\][/tex]
3. Rewrite the remaining expression:
[tex]\[
\frac{b^2 + 3b}{2b}
\][/tex]
4. Factor out [tex]\(b\)[/tex] from the numerator:
[tex]\[
\frac{b(b + 3)}{2b}
\][/tex]
5. Cancel [tex]\(b\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{b + 3}{2}
\][/tex]
So the simplified form of the given expression is:
[tex]\[
\frac{b + 3}{2}
\][/tex]
Therefore, the answer is [tex]\(\boxed{\frac{b+3}{2}}\)[/tex].
