jmc

An example of a quadratic function with zeroes at $x=2$ and $x=4$ is $(x-2)(x-4)$. However, when $x=3$, this function takes the value $-1$. However, multiplying the entire quadratic by $-6$ does not change the location of the zeroes, and does give us the desired value at $x=3$.

Thus, $-6(x-2)(x-4)$ has all the desired properties. The expanded form of this expression is $\boxed{-6x^2+36x-48}$.

Note that this is the only such quadratic. Any quadratic must factor as $a(x-r)(x-s)$, where its zeroes are $r$ and $s$; thus a quadratic with zeroes at $x=2$ and $x=4$ must be of the form $a(x-2)(x-4)$, and the coefficient $a=-6$ is forced by the value at $x=3$.