To solve the equation \(12 + 4x(X - 2) = 64\), let's first simplify and isolate the variable \(x\).

Starting with the equation:

\[
12 + 4x(X - 2) = 64
\]

First, subtract 12 from both sides:

\[
4x(X - 2) = 64 - 12
\]

This simplifies to:

\[
4x(X - 2) = 52
\]

Next, divide both sides by 4:

\[
x(X - 2) = \frac{52}{4}
\]

This gives:

\[
x(X - 2) = 13
\]

Now let's expand the left side:

\[
x^2 - 2x = 13
\]

Rearranging gives us a standard quadratic equation:

\[
x^2 - 2x - 13 = 0
\]

Next, we can use the quadratic formula to solve for \(x\):

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = 1\), \(b = -2\), and \(c = -13\). Plugging in these values:

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-13)}}{2 \cdot 1}
\]

This simplifies to:

\[
x = \frac{2 \pm \sqrt{4 + 52}}{2}
\]

\[
x = \frac{2 \pm \sqrt{56}}{2}
\]

Simplifying \(\sqrt{56}\):

\[
\sqrt{56} = \sqrt{4 \cdot 14} = 2\sqrt{14}
\]

So the expression for \(x\) becomes:

\[
x = \frac{2 \pm 2\sqrt{14}}{2}
\]

This reduces to:

\[
x = 1 \pm \sqrt{14}
\]

Thus, the two possible solutions are:

\[
x = 1 + \sqrt{14} \quad \text{and} \quad x = 1 - \sqrt{14}
\]

These are the solutions to the given equation.