Q.Find the value of $2x^4+5x^3+7x^2-x+41$, if $x = -2-\sqrt3\,i$.

Every quadratic equation $ax^2+bx+c=0$ (with $a\neq0$) has solutions given by the familiar formula $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$, and this formula continues to work even when the discriminant $D=b^2-4ac$ is negative or when the coefficients $a,b,c$ are themselves complex — the Fundamental Theorem of Algebra guarantees that a degree-$n$ polynomial equation always has exactly $n$ roots in the complex numbers, so a quadratic is never left without a solution. When $a,b,c$ are real and $D<0$, the square root $\sqrt{D}$ is rewritten as $i\sqrt{|D|}$, and the two roots that come out are complex conjugates of each other — if $p+iq$ is one root, $p-iq$ is automatically the other. When the coefficients are themselves complex (or when the discriminant is a general complex number rather than a negative real one), finding $\sqrt{D}$ requires the square-root-of-a-complex-number technique (writing the square root as $a+ib$ and solving two simultaneous equations), and the two r …