Question: show that 25x^2 + 20x + 12 has no integral zero.
I think that you mean, "the zeros of the function are not real"
Set the expression equal to zero.
25x^2 + 20x + 12 = 0
25x^2 + 20x + 4 + 8 = 0
(5x + 2)^2 + 8 = 0
(5x + 2)^2 = -8
5x + 2 = ± √(-8) = ± √(8)*√(-1) = ± 2√(2).i
5x + 2 = ± 2√(2).i
5x = -2 ± 2√(2).i
x = (-2 ± 2√(2).i)/5
The two solutions (zeros) of the expression are both complex values.
The solutions are: x = (-2 - 2√(2).i)/5, x = (2 + 2√(2).i)/5