Question

Why is -a² is always negative (a ≠ 0) and why is (-a)² is always positive?

Answer 1

$-a^2=(-1)\cdot a\cdot a$

Regardless of the sign of $a$, we have $a\cdot a=a^2\ge0$ (never negative). But multiplying by -1 makes it negative.

On the other hand,

$(-a)^2=((-1)\cdot a)^2=(-1)^2\cdot a^2=1\cdot a^2=a^2$

which can never be negative for real $a$.