**Given Information:**
a - b = 12
ab = 14

**To Find:**
a^2 - ab - b^2

**Solution:**

Let's start by multiplying the first equation (a - b = 12) by a:

a(a - b) = 12a
a^2 - ab = 12a

Now, we can substitute the value of ab from the second equation (ab = 14) into the equation above:

a^2 - ab = 12a
a^2 - 14 = 12a

Next, let's simplify this equation by moving all the terms to one side:

a^2 - 12a - 14 = 0

This is a quadratic equation. We can solve it by factoring or using the quadratic formula. Let's use the quadratic formula:

a = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a^2 - 12a - 14 = 0, the coefficients are:
a = 1, b = -12, c = -14

Plugging in these values, we get:

a = (-(-12) ± √((-12)^2 - 4(1)(-14))) / (2(1))
a = (12 ± √(144 + 56)) / 2
a = (12 ± √200) / 2
a = (12 ± 10√2) / 2
a = 6 ± 5√2

Therefore, we have two possible values for 'a':
a1 = 6 + 5√2
a2 = 6 - 5√2

Now, let's find the values of 'b' corresponding to each value of 'a':

For a1 = 6 + 5√2:
a1 - b = 12
(6 + 5√2) - b = 12
b = (6 + 5√2) - 12
b = -6 - 5√2

For a2 = 6 - 5√2:
a2 - b = 12
(6 - 5√2) - b = 12
b = (6 - 5√2) - 12
b = -6 + 5√2

Now, let's calculate a^2 - ab - b^2 for both values of 'a' and 'b':

For a1 = 6 + 5√2 and b = -6 - 5√2:
a1^2 - a1b - b^2 = (6 + 5√2)^2 - (6 + 5√2)(-6 - 5√2) - (-6 - 5√2)^2

Simplifying this expression, we get the value for a1^2 - ab - b^2.

Similarly, for a2 = 6 - 5√2 and b = -6 + 5√2, we can calculate a2^2 - a2b - b^2.

Please note that the final values of a^2 - ab - b^2 will depend on the exact calculations performed.

(a-b)^2 = a^2-b^2+2ab
(12)^2 = a^2-b^2+2(14)
144 = a^2-b^2+28
144-28 =a^2-b^2
116 = a^2-b^2