To find the solutions for the quadratic equation (n+20)x^2 + (n+5)x + 1 = 0, we can use the quadratic formula:

x = [-b ± √(b^2 - 4ac)] / 2a

where a = n+20, b = n+5, and c = 1.

Substitute these values into the formula:

x = [-(n+5) ± √((n+5)^2 - 4(n+20)(1))] / 2(n+20)

Expand the terms:

x = [-n-5 ± √(n^2 + 10n + 25 - 4n - 80)] / 2(n+20)
x = [-n-5 ± √(n^2 + 6n - 55)] / 2(n+20)

Since the expression under the square root sign is not a perfect square, the solutions are in terms of n:

x = (-n - 5 ± √(n^2 + 6n - 55)) / 2(n+20)