Question 140
Let $a$, $b$, $c$, $d$ be real numbers, with at least one of $a$ or $c$ non-zero, such that
$$
a^2 + ac + c^2 + 1 = b^2 + bd + d^2, \quad \text{and} \quad 2ab + ad + 2cd + bc = 0.
$$
Show that $ac \lt 0$ or $(3a^2 - b^2 + 1)(3c^2 - d^2 + 1) \le 0$.
$$
a^2 + ac + c^2 + 1 = b^2 + bd + d^2, \quad \text{and} \quad 2ab + ad + 2cd + bc = 0.
$$
Show that $ac \lt 0$ or $(3a^2 - b^2 + 1)(3c^2 - d^2 + 1) \le 0$.