To find the values of A and B, we need to use the fact that a, g are roots of the equation Ax^2 - 4x + 1 = 0, and b, d are roots of the equation Bx^2 - 6x + 1 = 0.

Since a and g are roots of Ax^2 - 4x + 1 = 0, we can substitute a and g into the equation to get two equations:

Aa^2 - 4a + 1 = 0 (1)
Ag^2 - 4g + 1 = 0 (2)

Similarly, since b and d are roots of Bx^2 - 6x + 1 = 0, we can substitute b and d into the equation to get two equations:

Bb^2 - 6b + 1 = 0 (3)
Bd^2 - 6d + 1 = 0 (4)

Now, let's solve these four equations to find the values of A and B.

From equation (1), we have:
Aa^2 - 4a + 1 = 0

From equation (2), we have:
Ag^2 - 4g + 1 = 0

Subtracting equation (1) from equation (2), we get:
Ag^2 - Aa^2 - 4g + 4a = 0

Factoring out a common factor of g - a, we have:
(A - 4)(g - a) = 0

Since a and g are distinct roots, we know that g - a ≠ 0, so we can divide both sides of the equation by (g - a):

A - 4 = 0

Therefore, A = 4.

Now, let's substitute this value of A into equation (1) to solve for a:

4a^2 - 4a + 1 = 0

This equation does not factor nicely, so we can use the quadratic formula to find the values of a:

a = (-(-4) ± sqrt((-4)^2 - 4(4)(1))) / (2(4))
a = (4 ± sqrt(16 - 16)) / 8
a = (4 ± sqrt(0)) / 8
a = 4/8
a = 1/2

Therefore, a = 1/2.

Now, let's solve equations (3) and (4) to find the value of B:

Bb^2 - 6b + 1 = 0 (3)
Bd^2 - 6d + 1 = 0 (4)

Subtracting equation (3) from equation (4), we get:
Bd^2 - Bb^2 - 6d + 6b = 0

Factoring out a common factor of d - b, we have:
(B - 6)(d - b) = 0

Since b and d are distinct roots, we know that d - b ≠ 0, so we can divide both sides of the equation by (d - b):

B - 6 = 0

Therefore, B = 6.

Now, let's substitute this value of B into equation (3) to solve for b:

6b^