Certainly! Let's go step-by-step to multiply these fractions and ultimately simplify the product.
We are given the following fractions to multiply:
1. [tex]\(\frac{q + 5}{2}\)[/tex]
2. [tex]\(\frac{4q}{q + 4}\)[/tex]
3. [tex]\(\frac{4q + 20q^2}{4q + 8}\)[/tex]
4. [tex]\(\frac{4q^2 + 20q}{2}\)[/tex]
5. [tex]\(\frac{q^2 + 20q}{2q}\)[/tex]
6. [tex]\(\frac{4q^2 + 20q}{2q + 8}\)[/tex]
Let's now multiply these fractions together:
[tex]\[
\frac{q + 5}{2} \cdot \frac{4q}{q + 4} \cdot \frac{4q + 20q^2}{4q + 8} \cdot \frac{4q^2 + 20q}{2} \cdot \frac{q^2 + 20q}{2q} \cdot \frac{4q^2 + 20q}{2q + 8}
\][/tex]
### Step 1: Combine the fractions
Combine the numerators and the denominators.
[tex]\[
\frac{(q + 5) \cdot 4q \cdot (4q + 20q^2) \cdot (4q^2 + 20q) \cdot (q^2 + 20q) \cdot (4q^2 + 20q)}{2 \cdot (q + 4) \cdot (4q + 8) \cdot 2 \cdot 2q \cdot (2q + 8)}
\][/tex]
### Step 2: Simplify each component
Let's now simplify each component individually:
- [tex]\(4q + 20q^2 = 4q(1 + 5q) = 4q(5q + 1)\)[/tex]
- [tex]\(4q + 8 = 4(q + 2)\)[/tex]
- [tex]\(4q^2 + 20q = 4q(q + 5)\)[/tex]
- [tex]\(q^2 + 20q = q(q + 20)\)[/tex]
- [tex]\(2q + 8 = 2(q + 4)\)[/tex]
### Step 3: Substitute back into fractions
Substituting these simplified forms back:
[tex]\[
\frac{(q + 5) \cdot 4q \cdot 4q(5q + 1) \cdot 4q(q + 5) \cdot q(q + 20) \cdot 4q(q + 5)}{2 \cdot (q + 4) \cdot 4(q + 2) \cdot 2 \cdot 2q \cdot 2(q + 4)}
\][/tex]
### Step 4: Combine and simplify the overall fraction
Combine the terms:
[tex]\[
\frac{4 \cdot 4 \cdot 4 \cdot 4 \cdot q \cdot q \cdot q \cdot q \cdot (q + 5) \cdot (q + 5) \cdot (5q + 1) \cdot (q + 5) \cdot (q + 20) \cdot (q + 5)}{2 \cdot 4 \cdot 4 \cdot 2 \cdot 2q \cdot (q + 4) \cdot (q + 2) \cdot (q + 4)}
\][/tex]
This simplifies to:
[tex]\[
\frac{4^4 \cdot q^4 \cdot (q + 5)^3 \cdot (5q + 1) \cdot (q + 20)}{2^4 \cdot q \cdot (q + 4)^2 \cdot (q + 2)}
\][/tex]
We know that [tex]\(4 = 2^2\)[/tex], so [tex]\(4^4 = (2^2)^4 = 2^8\)[/tex]:
[tex]\[
\frac{2^8 \cdot q^4 \cdot (q + 5)^3 \cdot (5q + 1) \cdot (q + 20)}{2^4 \cdot q \cdot (q + 4)^2 \cdot (q + 2)}
\][/tex]
Finally, cancel out common factors [tex]\(2^4\)[/tex]:
[tex]\[
\frac{2^{8-4} \cdot q^4 \cdot (q + 5)^3 \cdot (5q + 1) \cdot (q + 20)}{q \cdot (q + 4)^2 \cdot (q + 2)}
\][/tex]
### Final Simplified Form
[tex]\[
\frac{4q^4 (q + 5)^3 (5q + 1) (q + 20)}{(q + 2)(q + 4)^2}
\][/tex]
Thus, the simplified form of the product of the given fractions is:
[tex]\[
\frac{4q^4 (q + 5)^3 (q + 20)(5q + 1)}{(q + 2)(q + 4)^2}
\][/tex]
