Area enclosed by two curves, part 2

Calculus Level 5

The parabola $$f(x)=x^2$$ is tangent to the graph of $$g(x) = x^4 + ax^3 + \color{red}cx^2+ bx +1$$ at two distinct points. Given that the $${\color{purple}{minimum}}$$ area enclosed by these two curves is $$\frac{p}{q}$$, where $$p$$ and $$q$$ are coprime positive integers, find the value of $$p+q$$.

Remark: try the case for $$c=1$$ here

This problem is part of Curves... cut or touch?
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