# coefficient of Nonlinear relationship interpretation

I try to identify a nolinear relationship between a dependent variable and independent varaibles. In the literature, to detect this relation, we introduce the term. When I make a simple Regression ( OLS) with the introduction of variables (without their squared), the model gives not significant results, for the indépendents variables. On the other hand, it becomes significant, (similar for basic variables) when I add the square term. The problem that I do not know how interpret the sign of the variable coefficient in a non linear relationship.

Now, to your question: There are various ways that a quadratic relationship can mask a linear one. I'll give an example with only one IV, for simplicity. Suppose $Y = x^2 + e$ and x ranges from -10 to 10. Then trying to fit a linear relationship will result in a coefficient near 0 (and a p value near 1).
If $y = x^2 + x + e$ then it's possible to get a nonsignificant linear relationship that becomes significant when the quadratic term is added.