This is a variation of this question, but in my case I want to know if two polynomials of different degrees and non-negative coefficients can have more than one intersection in the positive $x$-axis.
For example, this is the plot of the polynomials $15 x^4$ and $45 x^3 + 105 x^2 + 85 x + 25$.
Of course they can. Say $3x$ and $x^2+2$. They intersect in $1$ and $2$:
$$ 3x=x^2+2\implies x^2-3x+2=0\implies (x-2)(x-1)=0$$