To solve the equation, we must first simplify each side separately before setting them equal to each other.

Starting with the left side, we have:
5^(4x+1) + 4 5^(2x) = 1
Rewrite 5^(2x) as (5^2)^x = 25^x
So, we can rewrite the equation as:
5^(4x+1) + 4 25^x = 1
Rewrite the equation using the property that (5^x)(5^y) = 5^(x+y)
5^4x 5^1 + 4 25^x = 1
5^4x 5 + 4 * 25^x = 1
5^(4x+1) + 100^x = 1 (1)

Now, moving on to the right side of the equation:
6^(1-x-2x^2) = 6^(2x^2 + x) - 5
Using the property that a^(m-n) = a^m / a^n
6^(1) / 6^x / 6^(2x^2) = 6^(2x^2 + x) - 5
6 / 6^x / 6^(2x^2) = 6^(2x^2 + x) - 5
6 / 6^x / 6^(2x^2) = 6^(2x^2) 6^x - 5
6 / 6^(2x^2) / 6^x = 36x^2 6^x - 5
6 / 36 6^x = 36x^2 6^x - 5
1 / 6^x = 6^2x - 5

Since, on the left side of the equation, 1 / 6^x = 1 / 6^x, we can rewrite equation (1) as:
1 / 5^(4x+1) + 100^x = 1 (2)

Now, we have two equations:
1 / 5^(4x+1) + 100^x = 1
1 / 5^(4x+1) + 100^x = 1

From the equations above, we see that both sides are equal to 1. Thus, we can conclude that this equation has no solution.