1. Solve the equation

$$
x^{3}-\left(m^{2}+3\right) x^{2}+\left(m^{2}+3\right) x+m^{4}-1=0
$$

for the unknown $x$ by factoring the polynomial $P(m)$ on the left side of this equation.
## Solution. We have

$$
P(m)=m^{4}-\left(x^{2}-x\right) m^{2}+(x-1)^{2}=\left(m^{2}-x+1\right)\left(m^{2}-(x-1)^{2}\right)
$$

The solutions to the given equation are:

$$
x_{1}=m^{2}+1, \quad x_{2}=m+1, \quad x_{3}=-m+1
$$