The value of $2x^4+5x^3+7x^2-x+41$ when $x=-2-\sqrt 3 i$ is

Question

Now I could solve this by just plugging in the given value of x in the expression and get the answer 6. What I am asking, is whether there is a shorter way to solve this, cause this took me a lot of time.

Thanks!

Answer 1

Notice that $(x+2)^2 = -3$ so $\boxed{x^2=-4x-7}$ and thus \begin{eqnarray}x^4 &=&(-4x-7)^2\\&=& 16x^2+56x+49\\&=&16(-4x-7)+56x+49\\&=& -64x-112+56x+49\\&=&-8x-63\end{eqnarray}

and $$x^3 = -4x^2-7x = +16x+28-7x = 9x+28$$

So $$...= 2(-8x-63)+5(9x+28)+7(-4x-7)-x+41=6$$