Solution
= Solution
{c}
Let $P=3x^4-5x^3+2x^2-3x+2$ and $Q=3x^3+x^2+x-2$. Applying <polynomial long division> gives the <euclidean algorithm>
$$
\begin{aligned}
P&=(x-2)Q+(3x^2+x-2),\\
Q&=x(3x^2+x-2)+(3x-2),\\
3x^2+x-2&=(x+1)(3x-2).
\end{aligned}
$$
Thus their <polynomial greatest common divisor> is $3x-2$, up to a nonzero constant factor. Their unique common <root of a polynomial> is therefore
$$
\boxed{\alpha=\frac23.}
$$