$$x+7 = \frac{x+1}{4x^4}$$
Answer
$$ \begin{matrix}x_1 = 0 & x_2 = 1.8746 & x_3 = -1.7701+0.9921i \\[1 em] x_4 = -1.7701-0.9921i & x_5 = 0.3328+1.87531i & x_6 = 0.3328-1.87531i \end{matrix} $$
Explanation
$$ \begin{aligned} x+7 &= \frac{x+1}{4x^4}&& \text{multiply ALL terms by } \color{blue}{ 4x^4 }. \\[1 em]4x^4x+4x^4\cdot7 &= 4x^4\frac{x+1}{4x^4}&& \text{cancel out the denominators} \\[1 em]4x^5+28x^4 &= x^9+x^8&& \text{move all terms to the left hand side } \\[1 em]4x^5+28x^4-x^9-x^8 &= 0&& \text{simplify left side} \\[1 em]-x^9-x^8+4x^5+28x^4 &= 0&& \\[1 em] \end{aligned} $$
In order to solve $ \color{blue}{ -x^{9}-x^{8}+4x^{5}+28x^{4} = 0 } $, first we need to factor our $ x^4 $.
$$ -x^{9}-x^{8}+4x^{5}+28x^{4} = x^4 \left( -x^{5}-x^{4}+4x+28 \right) $$$ x = 0 $ is a root of multiplicity $ 4 $.
The remaining roots can be found by solving equation $ -x^{5}-x^{4}+4x+28 = 0$.
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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