Find out value of $(3Z^{100}+\frac{2}{Z^{100}}+1)(Z^{100}+\frac{2}{Z^{100}}+3)?$

Question

If a complex number $Z$ satisfy following relationship $(Z+\frac{1}{Z})(Z+\frac{1}{Z}+1)=1$ then find out value of $(3Z^{100}+\frac{2}{Z^{100}}+1)(Z^{100}+\frac{2}{Z^{100}}+3)?$
My Attempt
let $u=(Z+\frac{1}{Z})(Z+\frac{1}{Z}+1)$ Then given condition become $(u)(u+1)=1$. $(3Z^{100}+\frac{2}{Z^{100}}+1)(Z^{100}+\frac{2}{Z^{100}}+3)=(2(Z^{100}+\frac{1}{Z^{100}})+Z^{100}+1)((Z^{100}+\frac{1}{Z^{100}})+\frac{1}{Z^{100}}+3)$. With this I expected to reduce it down in terms of $(Z+\frac{1}{Z})$ and then using given condition we can find the value. But I am unable to reduce it down in terms of $(Z+\frac{1}{Z})$. Can anyone help? Or am I making a mistake in strategy? Thanks in advance.

Answer 1

\begin{align*} & \left(z+\frac{1}{z}\right) \left(z+\frac{1}{z}+1\right) =1\\[6pt] \implies\;& \left(\frac{z^2+1}{z}\right) \left(\frac{z^2 + 1 + z}{z}\right) =1\\[4pt] \implies\;&(z^2+1)(z^2 + 1 + z)=z^2\\[4pt] \implies\;&z^4 + z^3 + 2z^2 + z +1= z^2\\[4pt] \implies\;&z^4 + z^3 + z^2 + z +1= 0\\[4pt] \implies\;&(z-1)(z^4 + z^3 + z^2 + z +1)= 0\\[4pt] \implies\;&z^5-1=0\\[4pt] \implies\;&z^5 = 1\\[4pt] \implies\;&z^{100}=1\\[4pt] \end{align*} The rest is easy.

Answer 2

HINT: from your condition we get $$z^2+2+\frac{1}{z^2}+z+\frac{1}{z}=1$$ or $$\left(z+\frac{1}{z}\right)^2+z+\frac{1}{z}=1$$