Let's first simplify the expression step by step:

Apply the distributive property to expand the terms:

(x-2)₂ - 5(x² - 9) + 4(x² + 3x - 4)

Simplify the expression further:

x - 2 - 5x² + 45 + 4x² + 12x - 16 = 1

Combine like terms:

-5x² + 4x² + x + 12x - 2 - 16 + 45 = 1

-x + 13x + 27 = 1

12x + 27 = 1

Subtract 27 from both sides:

12x = -26

Divide by 12:

x = -26/12
x = -13/6

Therefore, the solution to the equation (x-2)₂-5(x-3)(x+3)+4(x-1)(x+4) = 1 is x = -13/6.