Simplify \frac{1+\frac{2}{x+4}}{1+\frac{9}{x-3}} Step 1: Find the LCM of the denominators of the fractions in the numerator and denominator. Step 2: Multiply the numerator and denominator of the complex fraction by the LCM. Step 3: Factor \frac{x-3}{x+4} Step 4: Divide out common factors.

Given fraction is:
${\displaystyle \frac{1+\frac{2}{x+4}}{1+\frac{9}{x-3}}}$
To simplify the given fraction.
Step1. Find the LCM of the denominators of the fractions in numerators and denominators
${\displaystyle \frac{1+\frac{2}{x+4}}{1+\frac{9}{x-3}}=\frac{\frac{(x+4)+2}{(x+4)}}{\frac{(x-3)+9}{(x-3)}}}$
${\displaystyle \frac{\frac{(x+6)}{(x+4)}}{\frac{(x+6)}{x-3}}}$
Step 2. Multiply the numerator and denominator of the complex fraction by the LCM.
${\displaystyle \frac{\frac{(x+6)}{(x+4)}}{\frac{(x+6)}{x-3}}=\frac{(x+6)\times (x-3)}{(x+4)\times (x+6)}}$
Step 3. Factor
${\displaystyle \frac{(x+6)\times (x-3)}{(x+4)\times (x+6)}}$
Step 4. Divide out common factors
${\displaystyle \frac{(x+6)\times (x-3)}{(x+4)\times (x+6)}=\left(\frac{x-3}{x+4}\right)}$
Thus, the simplified form is ${\displaystyle \left(\frac{x-3}{x+4}\right)}$.