Finding the coefficient of $x^4$ of determinant

Question

Find the coefficient of $x^4$ in the following determinant.

$$\mspace{10mu}\begin{vmatrix} 3 & x & -1 &2 & 4\\ 1 & 2x & 1 & x &1 \\ -3 &2 & -1 & 1 &x \\ x& x & 4 & 5 &-2 \\ 2 &-3 &x & 4 &-5 \end{vmatrix}$$

I have no idea how to do it.

Answer 1

It seems to have found a solution. Not sure the simpliest one.

Let $f(x)=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5$ be the expending of determinant.

Then $\mspace{10mu}f(0)=a_0=78\mspace{10mu}$ , $\mspace{10mu}f(1)=a_0+a_1+a_2+a_3+a_4+a_5 \mspace{10mu}$ ,$\mspace{10mu}f(-1)=a_0-a_1+a_2-a_3+a_4-a_5$,

$f(2)=a_0+2a_1+4a_2+8a_3+16a_4+32a_5 \mspace{10mu},$

$f(-2)=a_0-2a_1+4a_2-8a_3+16a_4-32a_5\mspace{10mu}.$

Solving the linear system

$\{f(0)=a_0 ,f(1)+f(-1)=... ,f(2)+f(-2)=...\}$

we get $a_0,a_2,a_4.$

I hope there is more more convenient solution.