y = x2 + bx + 256 cuts the x axis at (h, 0) and (k, 0).

Explanatory Answer
The curve cuts the x axis at (h, 0) and (k, 0). So, h and k are the roots of the quadratic equation.
For a quadratic equation of the form ax2 + bx + c = 0, the sum of the roots = -b/a
The sum of the roots of this equation is -b/1 = -b.
Note: Higher the sum of the roots, lower the value of b.
For a quadratic equation of the form ax2 + bx + c = 0, the product of roots = c/a.
Therefore, the product of the roots of this equation = 256/1 = 256.
i.e., h * k = 256
h and k are both integers.
So, h and k are both integral factors of 256.
256 can be expressed as product of two numbers in the following ways:
1 * 256
2 * 128
4 * 64
8 * 32
16 * 16
The sum of the roots is maximum when the roots are 1 and 256 and the maximum sum is 1 + 256 = 257.
The least value possible for b is therefore -257.
The correct choice is (D)