Confidence Interval For Polynomial Fitting

I'm programming an n-dimensional polynomial fitting function. It uses the basic concept of least squares and a design matrix.

For example, a quadratic fit on 2d data:

This is a row in the design Matrix:

$X={({1, x, x^2})}$

And this is the matrix formula I'm using:

$(X^T*X)^{-1}(X^T*Y)=Ax^2+Bx+C$

This is nothing special or complicated.

I'm at the point however where I'd like to introduce confidence intervals. How do I compute the confidence intervals from this regression technique?