1. **Problem:** Given $f(x) = 2x^2 + 1$, find $g(x)$ where $g(x) = 2[f(x)]^2 - 1$. 2. **Formula:** To find $g(x)$, substitute $f(x)$ into the expression for $g(x)$: $$g(x) = 2[f(x)]^2 - 1$$ 3. **Step-by-step solution:** - Substitute $f(x)$: $$g(x) = 2(2x^2 + 1)^2 - 1$$ - Expand the square: $$(2x^2 + 1)^2 = (2x^2)^2 + 2 \cdot 2x^2 \cdot 1 + 1^2 = 4x^4 + 4x^2 + 1$$ - Substitute back: $$g(x) = 2(4x^4 + 4x^2 + 1) - 1$$ - Distribute 2: $$g(x) = 8x^4 + 8x^2 + 2 - 1$$ - Simplify: $$g(x) = 8x^4 + 8x^2 + 1$$ **Final answer:** $$g(x) = 8x^4 + 8x^2 + 1$$