Given Information:
- Real numbers: a, b, c, d
- a * b = 8
- a * b * c * d = 23
- a * d * b * c = 28
- c * d = 12

To Find:
The value of a * b * c * d.

Solution:

Step 1: Write the given information as equations.
- Equation 1: a * b = 8
- Equation 2: a * b * c * d = 23
- Equation 3: a * d * b * c = 28
- Equation 4: c * d = 12

Let's simplify the equations to make them easier to work with.

Step 2: Simplify Equation 3 by rearranging the terms.
- Equation 3: a * d * b * c = 28
- Rearranging the terms: a * b * c * d = 28

Now we have two equations with the same value for a * b * c * d.

Step 3: Compare Equation 2 and Equation 4.
- Equation 2: a * b * c * d = 23
- Equation 4: c * d = 12

We can substitute the value of c * d from Equation 4 into Equation 2.

Step 4: Substitute c * d = 12 into Equation 2.
- Equation 2: a * b * c * d = 23
- Substituting c * d = 12: a * b * 12 = 23

Now we have an equation with a * b as a variable.

Step 5: Solve for a * b.
- Equation: a * b * 12 = 23
- Dividing both sides by 12: a * b = 23/12

Step 6: Substitute the value of a * b into Equation 1.
- Equation 1: a * b = 8
- Substituting a * b = 23/12: 23/12 = 8

Now we have an equation with a * b as a variable.

Step 7: Solve for a * b.
- Equation: 23/12 = 8
- Multiplying both sides by 12: 23 = 96

This equation is not true, which means there is no solution for a, b, c, d that satisfies all the given conditions.

Conclusion: The value of a * b * c * d cannot be determined as there is no solution that satisfies all the given conditions.

A=b=c=4,d=3