-4x^2 + 9x - 1 = (x + 1)^2
-4x^2 + 9x - 1 = x^2 + 2x + 1
-4x^2 + 9x - 1 = x^2 + 2x + 1
Combine like terms:
-4x^2 + 9x - 1 = x^2 + 2x + 1
-5x^2 + 7x - 2 = 0

This is a quadratic equation that can be solved using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a

In this case:
a = -5, b = 7, c = -2

x = (-7 ± √(7^2 - 4(-5)(-2))) / 2*(-5)
x = (-7 ± √(49 - 40)) / -10
x = (-7 ± √9) / -10
x = (-7 ± 3) / -10

The two possible solutions are:
x = (-7 + 3) / -10 = -4 / -10 = 0.4
x = (-7 - 3) / -10 = -10 / -10 = 1

Therefore, the solutions to the equation are x = 0.4 and x = 1.