253. Find the necessary and sufficient condition for the equation

$$
x^{3}+a x^{2}+b x+c=0
$$

to be reduced to the form

$$
\left(x^{2}+m\right) \cdot(x+n)=0
$$

and indicate the simplest way to obtain the roots of the equation.

* See note on page 15.
According to the condition of the problem, the identity should hold:

$$
x^{3}+a x^{2}+b x+c=\left(x^{2}+m\right)(x+n)
$$

or

$$
x^{3}+a x^{2}+b x+c=x^{3}+m x+n x^{2}+m n
$$

from which

$$
m=b, n=a, m n=c
$$

or \( c=a b \) - this is the necessary condition. We will prove that this condition is also sufficient: if \( c=a b \), then

$$
x^{3}+a x^{2}+b x+a b=0
$$

or

$$
(x+a)\left(x^{2}+b\right)=0
$$

from which

$$
x_{1}=-a, x_{2,3}= \pm \sqrt{-b}
$$