To simplify the expression \((\frac{3x + 18}{4x + 8}) \div (\frac{x^2 - 36}{x + 2})\), we can follow these steps:
- Rewrite the division as multiplication by the reciprocal:
\[ \frac{3x + 18}{4x + 8} \cdot \frac{x + 2}{x^2 - 36} \]
- Factor all the polynomials involved:
- The numerator \(3x + 18\) can be factored as:
\[ 3(x + 6) \]
- The denominator \(4x + 8\) can be factored as:
\[ 4(x + 2) \]
- The expression \(x^2 - 36\) is a difference of squares and can be factored as:
\[ (x - 6)(x + 6) \]
Now substituting these factorizations into the expression, we have:
\[ \frac{3(x + 6)}{4(x + 2)} \cdot \frac{x + 2}{(x - 6)(x + 6)} \]
- Now multiply the fractions:
This becomes:
\[ \frac{3(x + 6)(x + 2)}{4(x + 2)(x - 6)(x + 6)} \]
- Cancel common factors:
The \(x + 2\) and \(x + 6\) terms in the numerator and denominator can be canceled out:
\[ \frac{3}{4(x - 6)} \]
So the simplified form of the given expression is:
\[ \frac{3}{4(x - 6)} \]
This is the simplest form of the expression.