1. Stated problem: Find the vertex coordinates of the quadratic function $f(x) = 2(x + 1)^2 + 1$. 2. Formula and rules: The vertex form of a quadratic function is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex. 3. Identify vertex: Here, $a = 2$, $h = -1$ (since $x + 1 = x - (-1)$), and $k = 1$. 4. Therefore, the vertex coordinates are $(-1, 1)$. Final answer: The vertex of the parabola is at $\boxed{(-1, 1)}$.