To solve the equation, we need to isolate the variable x. Let's start by simplifying the left side of the equation.

(5x - 6)³ + 6 = 10

First, let's remove the parentheses by cubing the expression (5x - 6).

(5x - 6)³ = (5x - 6)(5x - 6)(5x - 6)

Using the distributive property, we can expand this expression:

(5x - 6)(5x - 6)(5x - 6) = (25x² - 30x + 36)(5x - 6)

Now, let's multiply these two expressions:

(25x² - 30x + 36)(5x - 6) = 125x³ - 150x² + 180x - 150x² + 180x - 216

Combine like terms:

125x³ - 300x² + 360x - 216 = 10

Now, let's simplify the equation further:

125x³ - 300x² + 360x - 216 - 10 = 0

125x³ - 300x² + 360x - 226 = 0

Now, we have a cubic equation. To solve it, we can use factoring, the rational root theorem, or numerical methods. However, in this case, it seems that finding rational roots may be difficult. Therefore, we can use numerical methods or a graphing calculator to approximate the solutions.