Is it possible combine polynomial regression terms? I am dealing with a time series problem and one of the transformations that I am using is a moving slope of a line fit by linear regression on the last $w$ data, which is the window size.
However, my data is not always linear, so I thought of using a polynomial regression, but then I would have two terms representing what before was the slope:
$a + bx + cx^2$
Is there a way to combine $b$ and $c$?

No, you cannot combine them into one term.
If you want to know how much $y$ changes when $x$ changes by one unit, take the derivative. I’ll use partial derivative notation here so the solution makes sense for a regression with multiple different predictor variables ($x_1$ and $x_2$ instead of just $x$).
$$
\dfrac{\partial y}{\partial x} = b+2cx 
$$
The amount by which $y$ changes when $x$ changes by $1$ depends on the value of $x$, which should be consistent with how you think of the derivative of a parabola; the slope changes depending on where you examine the slope.
