1. **State the problem:** We have a quadratic function $g(x) = ax^2 + bx + c$ with roots at $x = -3$ and $x = -2$, and it passes through the point $(0, 12)$ on the y-axis. 2. **Use the roots to write the factored form:** Since the graph cuts the x-axis at $(-3, 0)$ and $(-2, 0)$, these are the roots of the quadratic. The factored form is: $$g(x) = a(x + 3)(x + 2)$$ where $a$ is a constant to be determined. 3. **Expand the factored form:** $$g(x) = a(x^2 + 5x + 6) = a x^2 + 5a x + 6a$$ So, comparing with $g(x) = ax^2 + bx + c$, we have: $$b = 5a, \quad c = 6a$$ 4. **Use the y-intercept to find $a$:** The graph passes through $(0, 12)$, so: $$g(0) = c = 12$$ From step 3, $c = 6a$, so: $$6a = 12$$ Divide both sides by 6: $$\cancel{6}a = \cancel{6}2$$ $$a = 2$$ 5. **Find $b$ and $c$ using $a=2$:** $$b = 5a = 5 \times 2 = 10$$ $$c = 6a = 6 \times 2 = 12$$ **Final answer:** $$a = 2, \quad b = 10, \quad c = 12$$