Consider the equation $(x^3-3x^2+4x-2)s(x)+(x^2-1)t(x)=x+1$. How can we find polynomials $s(x), t(x)$ which satisfy this equation?
Clearly $x+1$ divides the term involving $t(x)$ but not the first polynomial. If there are solutions then we need $(x+1)$ to divide $s(x)$. I am not sure where to go from here though.
hint
$$x^3-3x^2+4x-2=2(x-1)\Bigl(x^2-2x+2\Bigr)$$ Put $$s(x)=(x+1)R(x)$$ then
$$(x-1)\Bigl((x^2-2x+2)R(x)+t(x)\Bigr)=1$$ which is impossible.