To solve the equation \(8x^6 + 215x^3 - 27 = 0\), we can use substitution. Let \(y = x^3\). Then the equation becomes:

\(8y^2 + 215y - 27 = 0\)

We can factor this quadratic equation as:

\((8y - 1)(y + 27) = 0\)

Setting each factor to zero gives:

\(8y - 1 = 0 \quad \Rightarrow \quad y = \frac{1}{8}\)

\(y + 27 = 0 \quad \Rightarrow \quad y = -27\)

Since \(y = x^3\), we have:

\(x^3 = \frac{1}{8} \quad \Rightarrow \quad x = \frac{1}{2}\)

\(x^3 = -27 \quad \Rightarrow \quad x = -3\)

Thus, the solutions are \(x = \frac{1}{2}\) and \(x = -3\).