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I need to calculate the following determinant by using the properties of determinants.

$$\begin{vmatrix} x & 1 & x^2 + 4x - 2\\ -1 & x & x^2 - 4\\ 2 & -2 & x^2 - 2x + 4 \end{vmatrix}$$

I know that the result must be $(x^3 + x + 4)(x - 2)$, but I haven't figured out how to get it yet.

Thank you in advance!

George R.
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    You could solve this without using any properties but the definition. Are you asking for tricks to find the determinant in a way that is simpler that simply using the definition? – fuglede Nov 12 '16 at 12:02
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    Yes, this is what I want. – George R. Nov 12 '16 at 12:04
  • You can use Gauss-elimination or the "Entwicklungssatz" (I only know the german word) of Laplace. – Peter Nov 12 '16 at 12:06
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    Substituting $x=2$ I don't see at once that the determinant is zero and so I guess there are no obvious tricks, – lhf Nov 12 '16 at 12:14
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    @Peter, "Laplace expansion". – lhf Nov 12 '16 at 12:14

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