$\frac{3 g^{2}+12 g}{g+7} \cdot \frac{4}{8 g^{2}}$

Step 1 ：Simplify the expression inside the first fraction: \(\frac{3 g^{2}+12 g}{g+7}\)

Step 2 ：Simplify the expression inside the second fraction: \(\frac{4}{8 g^{2}}\)

Step 3 ：Multiply the two simplified fractions: \(\frac{3 g^{2}+12 g}{g+7} \cdot \frac{4}{8 g^{2}}\)

Step 4 ：Final Answer: \(\boxed{\frac{3(g + 4)}{2g(g + 7)}}\)