To multiply the two rational expressions \(\frac{2x + 14}{9x + 2} \cdot \frac{3x + 27}{x + 7}\), follow these steps:
1. Factor Simplification:
- First, factor the numerators and denominators of each fraction if possible.
For \(2x + 14\):
[tex]\[
2x + 14 = 2(x + 7)
\][/tex]
For \(3x + 27\):
[tex]\[
3x + 27 = 3(x + 9)
\][/tex]
So the expression becomes:
[tex]\[
\frac{2(x + 7)}{9x + 2} \cdot \frac{3(x + 9)}{x + 7}
\][/tex]
2. Cancel Common Factors:
- Identify and cancel out any common factors in the numerators and denominators.
Notice that \((x + 7)\) appears in both the numerator of the first fraction and the denominator of the second fraction, so they can be canceled. After canceling, the expression simplifies to:
[tex]\[
\frac{2 \cdot 3 (x + 9)}{9x + 2}
\][/tex]
3. Multiply the Remaining Terms:
- Multiply what's left of the numerators together and the denominators together:
[tex]\[
\frac{2 \cdot 3 (x + 9)}{9x + 2} = \frac{6(x + 9)}{9x + 2}
\][/tex]
Thus, the simplified form of the multiplication \(\frac{2x + 14}{9x + 2} \cdot \frac{3x + 27}{x + 7}\) is:
[tex]\[
\boxed{\frac{6(x + 9)}{9x + 2}}
\][/tex]
