How did the following factor out to this when doing PFE?

23 Views Asked by At

I'm currently applying PFE in getting the Z transform and this was the given:

\begin{equation} X(z) = \frac{z+3}{z^4+6z^3+14z^2+16z+8} \end{equation}

Dividing both sides by $z$ it resulted to this:

\begin{equation} \frac{X(z)}{z} = \frac{z+3}{z(z^4+6z^3+14z^2+16z+8)} \end{equation}

According to the book, factoring the denominator the result was: \begin{equation} \frac{X(z)}{z} = \frac{z+3}{z(z+2)^2(z+1-j)(z+1+j)} \end{equation}

With my own understanding I can get upto: \begin{equation} \frac{X(z)}{z} = \frac{z+3}{z(z+2)^2(z^2+2x+2)} \end{equation}

I need help in understanding how $z^2+2x+2$ factored into $(z+1-j)(z+1+j)$. I also tried doing long division and finding the complex roots but I just somehow couldn't get it.