Find all the real values of $\displaystyle a$ for which $\displaystyle a^3 + a^2|a + x| + |a^2x + 1| = 1$ has at least 4 integral solutions in $\displaystyle x$.
Find all the real values of $\displaystyle a$ for which $\displaystyle a^3 + a^2|a + x| + |a^2x + 1| = 1$ has at least 4 integral solutions in $\displaystyle x$.
Split this into four cases:
$\displaystyle a+x \ge 0,\ a^2x+1 \ge 0$
$\displaystyle a+x <0,\ a^2x+1 \ge 0$
$\displaystyle a+x \ge 0, \ a^2x+1 <0$
$\displaystyle a+x < 0, \ a^2x+1<0$
to give four problems but without absolute values in the equations.