1. **State the problem:** Simplify the expression $$\frac{8m^2 + 4m}{7(2m + 1)}$$. 2. **Identify the formula and rules:** We want to simplify a rational expression by factoring and canceling common factors. 3. **Factor the numerator:** $$8m^2 + 4m = 4m(2m + 1)$$. 4. **Rewrite the expression:** $$\frac{4m(2m + 1)}{7(2m + 1)}$$. 5. **Cancel the common factor $2m + 1$ (assuming $2m + 1 \neq 0$):** $$\frac{4m\cancel{(2m + 1)}}{7\cancel{(2m + 1)}} = \frac{4m}{7}$$. 6. **Final simplified form:** $$\frac{4m}{7}$$. This is the fully simplified expression.