To solve the equation \((x - 3)^5 = 4(x - 3)^3\), we can start by factoring out \((x - 3)^3\) from both sides.

1. Rewrite the equation:
\[
(x - 3)^5 - 4(x - 3)^3 = 0
\]

2. Factor out \((x - 3)^3\):
\[
(x - 3)^3 \left((x - 3)^2 - 4\right) = 0
\]

3. This gives us two factors to solve for:
- \((x - 3)^3 = 0\)
- \((x - 3)^2 - 4 = 0\)

### Solve \((x - 3)^3 = 0\):
\[
x - 3 = 0 \implies x = 3
\]

### Solve \((x - 3)^2 - 4 = 0\):
4. Rearranging gives:
\[
(x - 3)^2 = 4
\]

5. Taking the square root of both sides results in:
\[
x - 3 = 2 \quad \text{or} \quad x - 3 = -2
\]

6. Thus,
\[
x = 5 \quad \text{or} \quad x = 1
\]

### Summary of solutions:
The solutions to the equation \((x - 3)^5 = 4(x - 3)^3\) are:
\[
x = 3, \quad x = 5, \quad x = 1
\]