Enrique is constructing a doorway. The width of the doorway is framed from an 8-foot board with a length of 2√3

To find the area of the opening of the doorway that Enrique is constructing, we need to first calculate the dimensions of the doorway.

1. **Width of the Doorway:**
The width of the doorway is given as \(2\sqrt{3}\) feet which was cut from an 8-foot board. Therefore, the width \(W\) of the doorway is:
\[
W = 2\sqrt{3} \text{ feet}
\]

2. **Height of the Doorway:**
The length \(2\sqrt{3}\) feet that was cut from the first 8-foot board is then attached to another 8-foot board, which we use to frame the height of the door. The height \(H\) of the doorway is therefore:
\[
H = 2\sqrt{3} \text{ feet}
\]

3. **Area of the Opening:**
The area \(A\) of the opening of the doorway can be calculated using the formula for the area of a rectangle:
\[
A = \text{Width} \times \text{Height}
\]

Substituting in our values for \(W\) and \(H\):
\[
A = (2\sqrt{3}) \times (2\sqrt{3})
\]
\[
A = 4 \times 3
\]
\[
A = 12 \text{ square feet}
\]

Thus, the area of the opening of the doorway Enrique is building is \(\boxed{12}\) square feet.