Is the set of the solutions $x+y-(2a^2+2a)|z|=a^3-a$ a subspace?

Question

For the parameter $a\in \mathbb{R}$, determine whether the set of solutions of the equation

$$x+y-(2a^2+2a)|z|=a^3-a$$

is a subspace in $\mathbb{R^3}$ or not. Justify.

Answer 1

HINT

Let check the properties that a subspace has to fulfill.

Firstly check whether the solution $(x,y,z)=(0,0,0)$ is in the set that is

$$x+y-(2a^2+2a)|z|=a^3-a\implies a^3-a=a(a^2-1)=0 \implies a=0,\pm1$$