​ What does the fundamental theorem of algebra state about the equation 2x2+8x+14=0 ?

The following diagram illustrates the gist of the Fundamental Theorem of Algebra.

 

 \(P(x)=\underbrace{ax^n+bx^{n-1}+cx^{n-2}+...+yx+z}\\ \hspace{30mm}\text{n complex roots}\)

 

The degree of the polynomial of \(2x^2+8x+14\) is 2, so the number of complex roots is also 2. 

 

You can use the quadratic formula to find those roots.

 

\(2x^2+8x+14=0\)	Apply the quadratic formula!
\(x_{1,2}=\frac{-8\pm\sqrt{8^2-4*2*14}}{2*2}\)	Now it is a matter of simplifying.
\(x_{1,2}=\frac{-8\pm\sqrt{64-112}}{4}\)
\(x_{1,2}=\frac{-8\pm\sqrt{-48}}{4}\)
\(x_{1,2}=\frac{-8\pm\sqrt{48*-1}}{4}\)
\(x_{1,2}=\frac{-8\pm i\sqrt{48}}{4}\)
\(x_{1,2}=\frac{-8\pm i\sqrt{16*3}}{4}\)
\(x_{1,2}=\frac{-8\pm 4i\sqrt{3}}{4}\)
\(x_{1,2}=-2\pm i\sqrt{3}\)