Just imagine what the graph looks like. It starts above the $x$-axis, crosses below at $x=1$, is tangent to the $x$-axis at $x=2$ and $x=3$, and then crosses above at $x=4$, with an inflection point at $(4,0)$. Thinking about the shape, I count:
One inflection point between $x=1$ and $x=2$,
Two inflection points between $x=2$ and $x=3$,
Two inflection points between $x=3$ and $x=4$, and
One inflection point at $x=4$.
Thus there are six inflection points. This makes sense -- the second derivative should have eight zeroes, but two of them are at $x=3$, leaving six inflection points.