To solve this polynomial equation, we can use the Rational Root Theorem or trial and error to find the roots.

One of the simplest ways to find the roots is by trying different values of x that make the equation equal to zero.

Let's plug in x = 2:

4(2)^4 + 10(2)^3 - 125(2) - 54 = 0
4(16) + 10(8) - 250 - 54 = 0
64 + 80 - 250 - 54 = 0
144 - 304 = 0
-160 ≠ 0

Since x = 2 does not satisfy the equation, we try another value.

Let's try x = -2:

4(-2)^4 + 10(-2)^3 - 125(-2) - 54 = 0
4(16) + 10(-8) + 250 - 54 = 0
64 - 80 + 250 - 54 = 0
14 + 196 = 0
210 ≠ 0

Since x = -2 also does not satisfy the equation, we continue trying other values until we find the roots.