1. **State the problem:** We are given the quadratic function $$g(x) = 2x^2 + 4x - 5$$ and need to rewrite it in the form $$g(x) = a(x + h)^2 + k$$ where $$a, h, k \in \mathbb{Z}$$. 2. **Formula and rules:** The form $$a(x + h)^2 + k$$ is called the vertex form of a quadratic function. To convert from standard form $$ax^2 + bx + c$$ to vertex form, we complete the square. 3. **Complete the square:** Start with $$g(x) = 2x^2 + 4x - 5$$. Factor out the coefficient of $$x^2$$ from the first two terms: $$g(x) = 2(x^2 + 2x) - 5$$ 4. To complete the square inside the parentheses, take half of the coefficient of $$x$$, which is 2, half is 1, then square it: $$1^2 = 1$$. Add and subtract 1 inside the parentheses: $$g(x) = 2(x^2 + 2x + 1 - 1) - 5$$ Rewrite as: $$g(x) = 2((x + 1)^2 - 1) - 5$$ 5. Distribute the 2: $$g(x) = 2(x + 1)^2 - 2 - 5$$ Simplify constants: $$g(x) = 2(x + 1)^2 - 7$$ 6. **Final answer:** $$g(x) = 2(x + 1)^2 - 7$$ where $$a = 2$$, $$h = 1$$, and $$k = -7$$. This is the vertex form of the quadratic function.