To solve this equation, we will need to simplify both sides and then set them equal to each other.

Given equation: 1/5x^5 - cos(2x) = x^4 + 2sin(2x)

First, let's simplify the equation by expanding the trigonometric functions:

1/5x^5 - cos(2x) = x^4 + 2sin(2x)
= 1/5x^5 - cos(2x) = x^4 + 2(2sin(x)cos(x))
= 1/5x^5 - cos(2x) = x^4 + 4sin(x)cos(x)

Now, let's set up the equation and solve for x:

1/5x^5 - cos(2x) = x^4 + 4sin(x)cos(x)

Subtract x^4 from both sides:

1/5x^5 - x^4 - cos(2x) - 4sin(x)cos(x) = 0

Now use numerical methods or approximation techniques to find the value of x that satisfies this equation.