The discriminant must be non-negative: 6² - 4*1*k ≥ 0, thus k ≤ 9.

The vertex x-coordinate is given by -b/(2a) = -4/(2*-1) = 2.

The vertex x-coordinate is found using x = -b/(2a) = -6/(-2) = 3.

The minimum occurs at x = 2. f(2) = 2(2^2) - 8(2) + 10 = 2.

Set f'(x) = 0. f'(x) = 6x^2 - 18x + 12 = 0. Critical points are x = 2.

f'(x) = 6x^2 - 18x + 12. The critical points are x = 1 and x = 2. Test intervals show increase in (0, 3).

Find f'(x) = 6x^2 - 18x + 12, set to 0. Local maxima at x = 2 gives f(2) = 6.

The vertex occurs at x = 2. f(2) = 3(2^2) - 12(2) + 7 = -5.

The x-coordinate of the vertex is given by x = -b/(2a) = 12/(2*3) = 2.

The vertex is at x = -b/(2a) = 2. f(2) = 3.

The vertex occurs at x = -b/(2a) = 12/(2*3) = 2. f(2) = 3(2^2) - 12(2) + 9 = 3.

The vertex x-coordinate is found using -b/(2a) = 12/(2*3) = 2.

f'(x) = 2x + 2. Setting f'(x) = 0 gives x = -1. f(-1) = 1.

f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1, 3. f(2) = 1 is a local minimum.

Using the determinant formula, det(E) = 1*(1*0 - 4*6) - 2*(0*0 - 4*5) + 3*(0*6 - 1*5) = 1*(-24) - 2*(-20) - 15 = -24 + 40 - 15 = 1.

Determinant of E = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1.

Determinant of E = (1*4) - (2*2) = 4 - 4 = 0.

The discriminant must be non-negative: 2² - 4*1*k ≥ 0, which simplifies to k ≤ 1.

The discriminant must be less than zero: 6² - 4*1*k < 0, which simplifies to k > 9.

The discriminant must be non-negative: 6² - 4*1*k ≥ 0, thus k ≤ 9. The minimum value of k is -9.

The discriminant must be less than zero: k² - 4*1*9 < 0, thus k² < 36, so k > 6 or k < -6.

For both roots to be positive, k must be greater than 0 and k² > 4*9 = 36, thus k > 6.

For both roots to be negative, m must be greater than 0 and m² < 36. Thus, m must be in the range (-6, 0). The suitable value is -4.

The sum of the roots is 3 + 3 = 6, hence m = 6.

Using the sum of roots: 2 + (-2) = -p, hence p = 0.

For both roots to be negative, p must be greater than 0 and p² > 16. Thus, p < -4.

For both roots to be negative, p must be positive and p² > 4*9. Thus, p > 6, so p = -4 is valid.

The sum of the roots is 3 + 3 = 6, hence p = 6.

The first digit can be any of 10 digits, the second can be any of 9, the third can be any of 8, and the fourth can be any of 7. Total = 10 * 9 * 8 * 7 = 5040.

Each digit can be any of the 10 digits (0-9), so the total number of PINs is 10^4 = 10000.