Since (x-2) has a remainder of 1., we can also say that when x=2, P(X)=10. This because of a statement in the Polynomial remainder theorem.
Then, substituing x=2 and P(X)=10, we can isolate k:
[tex]\begin{gathered} 10=2^4-3(2)^2+k(2)-2 \\ 10=4+k(2)-2 \\ 10=2+k(2) \\ 8=k(2) \\ k=\frac{8}{2}=4 \end{gathered}[/tex]