I have two fourth-order polynomials $f(x)$ = $ax^4 + bx^3+cx^2+dx +e$ and $g(x)=a_1x^4 + b_1x^3+c_1x^2+d_1x +e_1$ and would like to find the angle between them. I have found the point of interaction between the two ($x_0,y_0$).
My question is now how do I proceed forward and obtain two straight lines so that I could subsequently find the angle between them?
Since you have already found the point of intersection, you need to find the angle made by the tangents to the polynomials at this point which can be found by differentiating $f$ and $g$ with respect to $x$. Finally, the acute angle between the tangents will be $$\theta=\arctan|\frac{f'(x_0)-g'(x_0)}{1+f'(x_0)g'(x_0)}|$$