How did 2x^2 + 7x - 25 = 11 even become 2x^2 + 7x - 25 - 24 = 24 - 24 in the first place? In your first equation, the right hand side is 11, but in your second equation, it has somehow became 24.
For your second part, (x + 7)(x - 7) is definitely not the solution. You are ignoring the additional 7x completely.
2x^2 + 7x - 25 = 11
2x^2 + 7x - 25 - 11 = 0
2x^2 + 7x - 36 = 0
(2x^2 + 7x) - 36 = 0
2(x^2 + (7/2)x) - 36 = 0
2(x + (7/4))^2 - 49/8 - 36 = 0
2(x + (7/4))^2 - 337/8 = 0
2(x + (7/4))^2 = 337/8
(x + (7/4))^2 = 337/16
x + 7/4 = sqrt(337/16) or = -sqrt(337/16)
x = -7/4 + sqrt(193/8) or x = -7/4 - sqrt(193/8)
Suppose you meant 2x^2 + 7x - 25 = 24 instead. We have:
2x^2 + 7x - 25 - 24 = 0
2x^2 + 7x - 49 = 0
(2x^2 + 7x) - 49 = 0
2(x^2 + (7/2)x) - 49 = 0
2(x + (7/4))^2 - 49/8 - 49 = 0
2(x + (7/4))^2 - 441/8 = 0
2(x + (7/4))^2 = 441 / 8
(x + (7/4))^2 = 441 / 16
x + 7/4 = sqrt(441/16) or = -sqrt(441/16)
x = -7/4 + sqrt(441/16) or x = -7/4 - sqrt(441/16)
x = -7/4 + (21/4) or x = -7/4 - (21/4)
x = 14/4 or x = -28/4
x = 7/2 or x = -7