To solve the equation \(4(x-2)^2 = (x-6)^2\), let's start by expanding both sides.

First, expand the left side:
\[ 4(x-2)^2 = 4(x^2 - 4x + 4) = 4x^2 - 16x + 16 \]

Next, expand the right side:
\[ (x-6)^2 = x^2 - 12x + 36 \]

Now, we have:
\[ 4x^2 - 16x + 16 = x^2 - 12x + 36 \]

Let's move all terms to one side to set the equation to zero:
\[ 4x^2 - 16x + 16 - x^2 + 12x - 36 = 0 \]

Combine like terms:
\[ 3x^2 - 4x - 20 = 0 \]

Now, we need to solve the quadratic equation \(3x^2 - 4x - 20 = 0\). We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = -4\), and \(c = -20\).

First, calculate the discriminant:
\[ b^2 - 4ac = (-4)^2 - 4(3)(-20) = 16 + 240 = 256 \]

Now, apply the quadratic formula:
\[ x = \frac{-(-4) \pm \sqrt{256}}{2(3)} = \frac{4 \pm 16}{6} \]

This gives us two solutions:
\[ x = \frac{4 + 16}{6} = \frac{20}{6} = \frac{10}{3} \]
\[ x = \frac{4 - 16}{6} = \frac{-12}{6} = -2 \]

So, the solutions to the equation \(4(x-2)^2 = (x-6)^2\) are:
\[ x = \frac{10}{3} \quad \text{and} \quad x = -2 \]