1. **State the problem:** Simplify the expression $$\frac{8k^2 + 2k}{7(4k + 1)}$$. 2. **Factor the numerator:** Factor out the common factor $2k$ from the numerator: $$8k^2 + 2k = 2k(4k + 1)$$ 3. **Rewrite the expression:** Substitute the factored numerator back into the expression: $$\frac{2k(4k + 1)}{7(4k + 1)}$$ 4. **Cancel common factors:** Since $4k + 1$ appears in both numerator and denominator, cancel it: $$\frac{2k\cancel{(4k + 1)}}{7\cancel{(4k + 1)}} = \frac{2k}{7}$$ 5. **Final simplified form:** The expression simplifies to: $$\boxed{\frac{2k}{7}}$$ This is the fully simplified form because no further common factors exist between numerator and denominator.