$\displaystyle f(x)=-4.9x^2+19.6x+58.8$

I'm a little confused as using the quadratic formula would leave me with $x=6$ and $x=-2$, but $(x-6)(x+2)$ does not equal the original function.

2. Multiplying out will give you $x^2-4x-12$ and if you multiply this by a factor of -4.9 you get the original f(x) you had.
You could also do this the other way, you are looking for roots to the equation f(x)=0 so you get $-4.9 x^2 +19.6x+58.8=0$. Divide through by 4.9 and you get the equation I gave above.

3. If we assume that f(x)=0

Then we assume that: $-4.9x^2+19.6x+58.8=0$
My $-4.9$ times table is a little rusty, so allow me to tidy this up a little:
$-49x^2+196x+588=0$
Times both sides by $-1$:
$49x^2-196x-588=0$
Divide both sides by $49$:
$x^2-4x-12=0$
$(x-6)(x+2)=0$
$x=6$ or $x=-2$

If you substitute those back into the original function, you will find that they give a solution of $f(x)=0$.

If $f(x)=0$, then $4.9f(x)=0$ too. Did you follow why?

Edit: Beaten again!