1. We are given two functions: $f(x) = -2x^2 + 2x - 1$ and $g(x) = 2x + 1$. We need to find $f(g(2))$, which means we first find $g(2)$ and then substitute that result into $f(x)$. 2. Calculate $g(2)$ using the formula for $g(x)$: $$g(2) = 2(2) + 1 = 4 + 1 = 5$$ 3. Now substitute $g(2) = 5$ into $f(x)$: $$f(5) = -2(5)^2 + 2(5) - 1$$ 4. Calculate the powers and multiply: $$f(5) = -2(25) + 10 - 1 = -50 + 10 - 1$$ 5. Simplify the expression: $$f(5) = -50 + 10 - 1 = -40 - 1 = -41$$ 6. Therefore, the value of $f(g(2))$ is: $$\boxed{-41}$$