1. **State the problem:** We want to express the quadratic expression $$q + 12x - qx^2$$ in the form $$a - b(x - c)^2$$ where $$a$$, $$b$$, and $$c$$ are expressed in terms of $$q$$. 2. **Recall the formula:** The expression $$a - b(x - c)^2$$ represents a quadratic in vertex form, where $$a$$ is the maximum value, $$b$$ is the coefficient controlling the width and direction, and $$c$$ is the x-coordinate of the vertex. 3. **Rewrite the original expression:** $$q + 12x - qx^2 = -qx^2 + 12x + q$$ 4. **Complete the square:** Start by factoring out $$-q$$ from the terms involving $$x$$: $$-q(x^2 - \frac{12}{q}x) + q$$ 5. **Complete the square inside the parentheses:** Take half the coefficient of $$x$$ inside the parentheses: $$\frac{12}{q} \times \frac{1}{2} = \frac{6}{q}$$ Square it: $$\left(\frac{6}{q}\right)^2 = \frac{36}{q^2}$$ Add and subtract this inside the parentheses: $$-q\left(x^2 - \frac{12}{q}x + \frac{36}{q^2} - \frac{36}{q^2}\right) + q$$ Rewrite as: $$-q\left(\left(x - \frac{6}{q}\right)^2 - \frac{36}{q^2}\right) + q$$ 6. **Distribute $$-q$$:** $$-q\left(x - \frac{6}{q}\right)^2 + q \times \frac{36}{q^2} + q = -q\left(x - \frac{6}{q}\right)^2 + \frac{36}{q} + q$$ 7. **Combine constants:** $$a = q + \frac{36}{q}$$ $$b = q$$ $$c = \frac{6}{q}$$ 8. **Final expression:** $$q + 12x - qx^2 = a - b(x - c)^2 = \left(q + \frac{36}{q}\right) - q\left(x - \frac{6}{q}\right)^2$$ **Answer:** $$a = q + \frac{36}{q}, \quad b = q, \quad c = \frac{6}{q}$$