1. The problem is to find the value of the function $hgf(x)$ at $x=3$. 2. The function is given as: $$hgf(x) = \frac{4x + 12}{x + 8} + 3$$ 3. To simplify, combine the terms over a common denominator: $$hgf(x) = \frac{4x + 12}{x + 8} + \frac{3(x + 8)}{x + 8} = \frac{4x + 12 + 3x + 24}{x + 8} = \frac{7x + 36}{x + 8}$$ 4. Now substitute $x=3$ into the simplified function: $$hgf(3) = \frac{7(3) + 36}{3 + 8} = \frac{21 + 36}{11} = \frac{57}{11}$$ 5. Convert the improper fraction to a mixed number: $$\frac{57}{11} = 5 \frac{2}{11}$$ 6. Therefore, the value of $hgf(3)$ is $5 \frac{2}{11}$.