Quotient and remainder of $(X^4 -7X+1)$ modulo $(2X^2+1)$ in $\mathbb Z_7[X]$

Question

I am not sure if I have understood how to do this problem correctly.

Compute the quotient and remainder of

$(X^4 -7X+1)\ modulo\ (2X^2+1)\ in\ \mathbb Z_7[X]$

I got the quotient to be $4x^2 -2$ and the remainder be 3, is this correct?

Answer 1

Your answer is correct because $(2x^2+1)(4x^2-2)+3 = 8 x^4 + 1 = x^4 -7x+1$ in $\mathbb Z_7[x]$.