Nonlinear polynomial Diophantine equations

Question

I am interested in whether there exist nontrivial (i.e., non-constant) solutions to any or all of the equations \begin{align*} P(x, x^{-1})^2 + Q(x, x^{-1})^2 + 1 &= 0, \\ P(x, x^{-1})^2 + Q(x, x^{-1})^4 + 1 &= 0, \\ P(x, x^{-1})^2 + Q(x, x^{-1})^3 + 1 &= 0, \\ P(x, x^{-1})^2 + Q(x, x^{-1})^3 + Q(x, x^{-1}) &= 0, \\ P(x, x^{-1})^2 + Q(x, x^{-1})^5 + 1 &= 0, \end{align*} where $P$ and $Q$ are single-variable Laurent polynomials with coefficients in $\mathbb{C}$. For example, a straightforward solution to the first equation is \begin{equation*} P(x, x^{-1}) = \frac{x - x^{-1}}{2}, \quad Q(x, x^{-1}) = \frac{i(x + x^{-1})}{2}. \end{equation*} I would like to know whether the other equations have nontrivial solutions, and if so, whether there exists a method for generating solutions or for classifying all solutions. Comments on the more general equation $P(x, x^{-1})^a + Q(x, x^{-1})^b + 1 = 0$, where $a$ and $b$ are positive integers, would be appreciated as well.