To solve the equation, we can simplify it first.

We have 4^x * 4^3 - 4^x = 1008.

Using the properties of exponents, we can simplify the left side of the equation:

4^x * 4^3 - 4^x = 4^(x+3) - 4^x.

Now, we can rewrite the equation as:

4^(x+3) - 4^x = 1008.

To solve for x, we can set up a quadratic equation:

4^(x+3) - 4^x - 1008 = 0.

Let's substitute 4^x with a variable, say y:

4^3 * y - y - 1008 = 0.

Simplifying further:

64y - y - 1008 = 0.

Combining like terms:

63y - 1008 = 0.

Adding 1008 to both sides:

63y = 1008.

Dividing both sides by 63:

y = 16.

Now, substitute y back into the equation:

4^x = 16.

To solve for x, we can take the logarithm of both sides:

log(4^x) = log(16).

Using the logarithmic property, we can bring down the exponent:

x * log(4) = log(16).

Dividing both sides by log(4):

x = log(16) / log(4).

Using a calculator, we can evaluate this expression:

x ≈ 2.

Therefore, the solution to the equation 4^x * 4^3 - 4^x = 1008 is x ≈ 2.