1. Statement of the problem. Problem: Determine the truth value of $\forall x\,\forall y\,(x^2 + y^2 \ge -1)$ where $x,y$ are real numbers. 2. Formula and important rules used. We use the rule that for every real number $t$ we have $t^2 \ge 0$. Also the sum of nonnegative numbers is nonnegative, i.e., if $a \ge 0$ and $b \ge 0$ then $a + b \ge 0$. 3. Intermediate work and simplification. For any real $x$ we have $x^2 \ge 0$. For any real $y$ we have $y^2 \ge 0$. Therefore $x^2 + y^2 \ge 0$. Since $0 \ge -1$, we conclude $x^2 + y^2 \ge -1$. 4. Final conclusion and truth value. Because the inequality holds for arbitrary real $x$ and $y$, the quantified statement $\forall x\,\forall y\,(x^2 + y^2 \ge -1)$ is true. Final answer: True.