factor

x² - 48x - 2 = (x + a)(x +b)

 

a + b = -48 → b = -48 - a

ab = -2

 

a(-48 - a) = -2

-48a - a² = -2

a² + 48a = 2

a² + 48a + 576 = 2 + 576

(a + 24)² = 578

a = sqrt(578) - 24 = 17sqrt(2) - 24

b = -17sqrt(2) - 24

 

(x + 17sqrt(2) - 24)(x - 17sqrt(2) - 24)