1. **State the problem:** We are given the base and height of a triangle as polynomials in $q$: - Base = $2q + 2$ - Height = $5q + 4$ We need to: a. Write an expression for the area of the triangle. b. Calculate the area when $q = \frac{1}{2}$. 2. **Formula for the area of a triangle:** $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. **Write the expression for the area:** $$\text{Area} = \frac{1}{2} (2q + 2)(5q + 4)$$ 4. **Multiply the polynomials:** $$(2q + 2)(5q + 4) = 2q \times 5q + 2q \times 4 + 2 \times 5q + 2 \times 4$$ $$= 10q^2 + 8q + 10q + 8$$ $$= 10q^2 + 18q + 8$$ 5. **Substitute back into the area formula:** $$\text{Area} = \frac{1}{2} (10q^2 + 18q + 8)$$ 6. **Distribute $\frac{1}{2}$:** $$\text{Area} = \frac{1}{2} \times 10q^2 + \frac{1}{2} \times 18q + \frac{1}{2} \times 8$$ $$= 5q^2 + 9q + 4$$ 7. **Calculate the area when $q = \frac{1}{2}$:** $$\text{Area} = 5\left(\frac{1}{2}\right)^2 + 9\left(\frac{1}{2}\right) + 4$$ $$= 5 \times \frac{1}{4} + 9 \times \frac{1}{2} + 4$$ $$= \frac{5}{4} + \frac{9}{2} + 4$$ 8. **Find common denominator and add:** $$\frac{5}{4} + \frac{9}{2} + 4 = \frac{5}{4} + \frac{18}{4} + \frac{16}{4} = \frac{5 + 18 + 16}{4} = \frac{39}{4}$$ 9. **Final answer:** $$\boxed{\text{Area} = 5q^2 + 9q + 4}$$ $$\boxed{\text{Area when } q=\frac{1}{2} = \frac{39}{4} = 9.75}$$