1. **State the problem:** Divide the expression $\frac{q^2 + 2q}{q + 2}$ by $q$ and write the answer in simplest form. 2. **Rewrite the division as multiplication by the reciprocal:** $$\frac{q^2 + 2q}{q + 2} \div q = \frac{q^2 + 2q}{q + 2} \times \frac{1}{q}$$ 3. **Combine the fractions:** $$\frac{q^2 + 2q}{q + 2} \times \frac{1}{q} = \frac{q^2 + 2q}{q(q + 2)}$$ 4. **Factor the numerator:** $$q^2 + 2q = q(q + 2)$$ 5. **Substitute the factored form:** $$\frac{q(q + 2)}{q(q + 2)}$$ 6. **Cancel common factors:** $$\frac{\cancel{q}(\cancel{q + 2})}{\cancel{q}(\cancel{q + 2})} = 1$$ 7. **Final answer:** $$1$$ The expression simplifies to 1.