To solve the equation \(3(x-6)^2 = 60 - 10x\), follow these steps:

1. **Expand and simplify the equation:**
\[
3(x-6)^2 = 60 - 10x
\]

2. **Expand \((x-6)^2\):**
\[
(x-6)^2 = x^2 - 12x + 36
\]

3. **Substitute back into the equation:**
\[
3(x^2 - 12x + 36) = 60 - 10x
\]

4. **Distribute the 3:**
\[
3x^2 - 36x + 108 = 60 - 10x
\]

5. **Move all terms to one side to set the equation to 0:**
\[
3x^2 - 36x + 108 - 60 + 10x = 0
\]

6. **Combine like terms:**
\[
3x^2 - 26x + 48 = 0
\]

7. **Solve the quadratic equation \(3x^2 - 26x + 48 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = -26\), and \(c = 48\):**

\[
x = \frac{-(-26) \pm \sqrt{(-26)^2 - 4 \cdot 3 \cdot 48}}{2 \cdot 3}
\]

8. **Simplify inside the square root:**
\[
x = \frac{26 \pm \sqrt{676 - 576}}{6}
\]

9. **Simplify further:**
\[
x = \frac{26 \pm \sqrt{100}}{6}
\]

10. **Take the square root of 100:**
\[
x = \frac{26 \pm 10}{6}
\]

11. **Solve for the two possible values of \(x\):**
\[
x = \frac{26 + 10}{6} = \frac{36}{6} = 6
\]
\[
x = \frac{26 - 10}{6} = \frac{16}{6} = \frac{8}{3}
\]

So, the solutions to the equation \(3(x-6)^2 = 60 - 10x\) are:
\[
x = 6 \quad \text{and} \quad x = \frac{8}{3}
\]