Why is this QCQP non-convex?

Why is the following QCQP non-convex?

$$\begin{array}{ll} \underset{x,y \in \Bbb R}{\text{minimize}} & (x+2)^2 + (y+2)^2\\ \text{subject to} & x y \ge 4\\ & x \le 4\\ & y \ge 2\end{array}$$

The objective function is convex and the set of feasible solutions is also convex (follows from the graph).

The problem is not convex because it is not written as $\min \{ f(x) : g_i(x) \leq 0, i=1,2,\ldots,m \}$ where $f$ and $g_i$ are convex functions.

The feasible region is convex as you point out. You can obtain a convex formulation by applying a logarithmic transformation to the nonconvex constraint: $\log x + \log y \geq \log 4$. This constraint implicitly restricts $x$ and $y$ to positive reals, which looking at the other constraints does not reduce the feasible region.

Obviously, the objective function and the last two constraints are convex (quadratic, linear, linear respectively). If we look at the first constraint and transform it into the standard form as $-xy+4\leq0.$

The Hessian is

$\begin{bmatrix}0 & -1\\-1 & 0\end{bmatrix},$

which gives eigenvalues of -1 and 1, thus the Hessian is not positive semi-definite and the constraint is not convex. Combining all these facts, the optimization problem is not convex.