Let's break down and solve the given equation step by step.

The equation is:

\[
(3x - 2)^3 - 14 = 2 \times 5^2
\]

First, simplify the right-hand side:

\[
5^2 = 25
\]

So the equation becomes:

\[
(3x - 2)^3 - 14 = 2 \times 25
\]

Now, simplify the multiplication on the right-hand side:

\[
2 \times 25 = 50
\]

Thus, the equation is now:

\[
(3x - 2)^3 - 14 = 50
\]

Next, add 14 to both sides to isolate the cube term:

\[
(3x - 2)^3 = 50 + 14
\]

Simplify the right-hand side:

\[
(3x - 2)^3 = 64
\]

Now, take the cube root of both sides to solve for \(3x - 2\):

\[
3x - 2 = \sqrt[3]{64}
\]

Since \(64 = 4^3\), the cube root of 64 is 4:

\[
3x - 2 = 4
\]

Add 2 to both sides to solve for \(3x\):

\[
3x = 4 + 2
\]

Simplify the right-hand side:

\[
3x = 6
\]

Finally, divide both sides by 3 to solve for \(x\):

\[
x = \frac{6}{3}
\]

Simplify:

\[
x = 2
\]

So, the solution to the equation is:

\[
x = 2
\]