Number of real values of x that satisfy this determinant?

Question

How many real values of $x$ satisfy the following determinant?

$$ \begin{vmatrix} x & 3x+2 & 2x-1 \\ 2x-1 & 4x & 3x+1 \\ 7x-2 & 17x+6 & 12x-1 \\ \end{vmatrix} = 0 $$

If we expand the determinant, it should give us an equation of the third degree. So, there should be either three or one real solutions for it, as imaginary solutions only exist in pairs. Is my reasoning correct? And how do I find the exact number of solutions?

Do there exist both objective amd subjective solutions for this question? I'd like to know both.

Thank you so much!

Answer 1

Denoting by $L_i$ the $i-$th row, you have $$L_3 = 2L_2 + 3L_1$$

so the determinant is $0$ for all values of $x$.

Answer 2

Actually, that determinant is $0$, for each real number $x$.