Let $F:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be a real-valued polynomial function of the form $F(x,y)={\sum}_{i=\mathrm{0}}^{s}\mathrm{\u200d}{f}_{i}(x){y}^{i}$, with degree of $\mathrm{}y\mathrm{}\mathrm{}$ in $F(x,y)=s\ge \mathrm{}\mathrm{}\mathrm{1},\mathrm{}\mathrm{}x\in \mathbb{R}\mathbb{}.$ An irreducible real-valued polynomial function $p(x)$ and a nonnegative integer $m$ are given to find a polynomial function $y(x)\in \mathbb{R}[x]$ satisfying the following expression: $F(x,y(x))=c{p}^{m}(x)$ for some constant $c\in \mathbb{R}$. The constant $c$ is dependent on the solution $y(x)$, namely, a quasi-fixed (polynomial) solution of the polynomial-like equation $(\mathrm{*})$. In this paper, we will provide a non-NP-complete algorithm to solve all quasi-fixed solutions if the equation $(\mathrm{*})$ has only a finite number of quasi-fixed solutions.

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