Let $a$ be a real number. Consider the quadratic expressions in $x$ $$\begin{aligned}
& A = x^2 + ax + 1 \\
& B = x^2 + (a+3)x + 4
\end{aligned}$$ (1) The range of values taken by $a$ such that there exists a real number $x$ satisfying $A + B = 0$ is $$a \leq -\sqrt{\mathbf{AB}} - \frac{\mathbf{C}}{\mathbf{D}} \text{ or } \sqrt{\mathbf{AB}} - \frac{\mathbf{C}}{\mathbf{D}} \leq a.$$ (2) The range of values taken by $a$ such that there exists a real number $x$ satisfying $AB = 0$ is $$a \leq \mathbf{EF} \text{ or } \mathbf{G} \leq a.$$ (3) There exists a real number $x$ satisfying $A^2 + B^2 = 0$ only when $a = \mathbf{H}$. In this case $x = \mathbf{IJ}$.
Let $a$ be a real number. Consider the quadratic expressions in $x$
$$\begin{aligned}
& A = x^2 + ax + 1 \\
& B = x^2 + (a+3)x + 4
\end{aligned}$$
(1) The range of values taken by $a$ such that there exists a real number $x$ satisfying $A + B = 0$ is
$$a \leq -\sqrt{\mathbf{AB}} - \frac{\mathbf{C}}{\mathbf{D}} \text{ or } \sqrt{\mathbf{AB}} - \frac{\mathbf{C}}{\mathbf{D}} \leq a.$$
(2) The range of values taken by $a$ such that there exists a real number $x$ satisfying $AB = 0$ is
$$a \leq \mathbf{EF} \text{ or } \mathbf{G} \leq a.$$
(3) There exists a real number $x$ satisfying $A^2 + B^2 = 0$ only when $a = \mathbf{H}$. In this case $x = \mathbf{IJ}$.