To solve the equation \((4x^2 - x)(x^2 - 25) = 0\), we can use the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for \(x\).

1. Set the first factor equal to zero:
\[
4x^2 - x = 0
\]

Factor out \(x\):
\[
x(4x - 1) = 0
\]

This gives us two solutions:
\[
x = 0 \quad \text{or} \quad 4x - 1 = 0
\]

Solve \(4x - 1 = 0\):
\[
4x = 1
\]
\[
x = \frac{1}{4}
\]

So, the solutions from the first factor are:
\[
x = 0 \quad \text{or} \quad x = \frac{1}{4}
\]

2. Set the second factor equal to zero:
\[
x^2 - 25 = 0
\]

This is a difference of squares, which can be factored as:
\[
(x - 5)(x + 5) = 0
\]

This gives us two solutions:
\[
x - 5 = 0 \quad \text{or} \quad x + 5 = 0
\]

Solve these equations:
\[
x = 5 \quad \text{or} \quad x = -5
\]

So, the solutions from the second factor are:
\[
x = 5 \quad \text{or} \quad x = -5
\]

Combining all the solutions, we have:
\[
x = 0, \quad x = \frac{1}{4}, \quad x = 5, \quad x = -5
\]

Therefore, the complete set of solutions to the equation \((4x^2 - x)(x^2 - 25) = 0\) is:
\[
x = 0, \quad x = \frac{1}{4}, \quad x = 5, \quad x = -5
\]