1. **State the problem:** We need to rewrite the quadratic function $$g(x) = 5x^2 + 20x - 12$$ in the form $$g(x) = a(x + h)^2 + k$$ where $$a, h, k \in \mathbb{Z}$$. 2. **Recall the formula:** The form $$a(x + h)^2 + k$$ is called the vertex form of a quadratic function. It is useful because it shows the vertex of the parabola directly. 3. **Step 1: Factor out the coefficient of $$x^2$$ from the first two terms:** $$g(x) = 5(x^2 + 4x) - 12$$ 4. **Step 2: Complete the square inside the parentheses:** Take half of the coefficient of $$x$$, which is 4, so half is 2, then square it: $$2^2 = 4$$. Add and subtract 4 inside the parentheses to complete the square: $$g(x) = 5(x^2 + 4x + 4 - 4) - 12$$ 5. **Step 3: Group the perfect square trinomial and simplify:** $$g(x) = 5((x + 2)^2 - 4) - 12$$ 6. **Step 4: Distribute the 5 and simplify constants:** $$g(x) = 5(x + 2)^2 - 5 \times 4 - 12$$ $$g(x) = 5(x + 2)^2 - 20 - 12$$ $$g(x) = 5(x + 2)^2 - 32$$ 7. **Final answer:** $$g(x) = 5(x + 2)^2 - 32$$ This is the vertex form with $$a=5$$, $$h=2$$, and $$k=-32$$. This form shows the parabola opens upwards (since $$a=5>0$$) and has its vertex at $$(-2, -32)$$.