Đáp án:
$\begin{array}{l}
{a^3} - 2{a^2} + 3a + 1\\
= {a^3} + 3a - 2{a^2} - 6 + 7\\
= a\left( {{a^2} + 3} \right) - 2\left( {{a^2} + 3} \right) + 7\\
= \left( {{a^2} + 3} \right)\left( {a - 2} \right) + 7\\
Do:\left( {{a^2} + 3} \right)\left( {a - 2} \right) \vdots \left( {{a^2} + 3} \right)\\
\Rightarrow 7 \vdots \left( {{a^2} + 3} \right)\\
\Rightarrow {a^2} + 3 = 7\left( {do:{a^2} + 3 \ge 3} \right)\\
\Rightarrow {a^2} = 4\\
\Rightarrow \left[ \begin{array}{l}
a = 2\\
a = - 2
\end{array} \right.\\
Vậy\,a = 2;a = - 2
\end{array}$
