$$(2x+1)^3+7 = \frac{8}{(2+1)^3}$$
Answer
$$ \begin{matrix}x_1 = -1.44278 & x_2 = -0.02861+0.81647i & x_3 = -0.02861-0.81647i \end{matrix} $$
Explanation
$$ \begin{aligned} (2x+1)^3+7 &= \frac{8}{(2+1)^3}&& \text{multiply ALL terms by } \color{blue}{ 27 }. \\[1 em]27(2x+1)^3+27\cdot7 &= 27\cdot\frac{8}{(2+1)^3}&& \text{cancel out the denominators} \\[1 em]27(2x+1)^3+189 &= 8&& \text{simplify left side} \\[1 em]27(8x^3+12x^2+6x+1)+189 &= 8&& \\[1 em]216x^3+324x^2+162x+27+189 &= 8&& \\[1 em]216x^3+324x^2+162x+216 &= 8&& \text{move all terms to the left hand side } \\[1 em]216x^3+324x^2+162x+216-8 &= 0&& \text{simplify left side} \\[1 em]216x^3+324x^2+162x+208 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using qubic formulas.
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Equations Solver