12. Find all values of the coefficients $a, b$, and $c$ of the polynomial

$$
f(x)=x^{4}-x^{3}+a x^{2}+b x+c
$$

if it is known that it is divisible by the polynomial $\varphi(x)=x^{3}-2 x^{2}-5 x+6$ without a remainder.
$\triangle$ Let's divide the first polynomial by the second with a remainder:

$$
\begin{gathered}
x^{4}-x^{3}+a x^{2}+b x+c=\left(x^{3}-2 x^{2}-5 x+6\right)(x+1)+ \\
+(a+7) x^{2}+(b-1) x+(c-6)
\end{gathered}
$$

The remainder must be identically equal to zero:

$$
(a+7) x^{2}+(b-1) x+(c-6) \equiv 0
$$

Since two polynomials are identical if and only if they consist of the same terms (Theorem 1), we have:

$$
a+7=0, \quad b-1=0, \quad c-6=0
$$

From this, we find $a, b$, and $c$.

Answer: $a=-7, b=1, c=6$.