Solution:

Answer: 7. Because the first curve touches the $x$-axis at $x=0$ and $x=3$ while the second curve crosses the $x$-axis at $x= \pm 1$ and $x=2$, there are four points of intersection. In particular, the points of intersection have $x$-coordinates determined by the difference of the two curves:
$$0 = x^{2}(x-3)^{2} - (x^{2}-1)(x-2) = (x^{4} - 6x^{3} + \cdots) - (x^{3} + \cdots) = x^{4} - 7x^{3} + \cdots$$
We need only the first two coefficients to determine $x_{1} + x_{2} + x_{3} + x_{4} = -\left(\frac{-7}{1}\right) = 7$.