There is a formula :

Force = Density * Area * Speed^2

I get how we get from f=ma to:

F=(mv-mu)/t
F=(m(change in velocity))/t
F=(m/t)*(change in velocity)
F=((Density*volume)*(change in velocity))/time
But then from the last listed equation how do we get to
Force = Density * Area * Speed^2

There is a formula :

Force = Density * Area * Speed^2

Can you tell us where you found this formula? This looks like a formula for drag felt by an object moving through a fluid (or air), though usually it includes a constant that represents the drag coefficient of the shape of the object, and also a factor of 1/2:

F_d = C_d \frac 1 2 \rho A v^2

Here C_d is the drag coefficient, rho is the density of the fluid (not the density of the object moving through it), and A is the frontal cross-section of the object. Is this the equation you are thinking of? If so, it not derived from F=ma, but rather from energy principles. As the object moves through the fluid KE is imparted onto the fluid to move it aside, and this means there is a force between the object and the fluid, which we think of as drag. If the object has a blunt shape (like a cube) such that C_d=1, then from energy principles:

\frac 1 2 m v^2 = F d

where 'd' is the length of the object. The mass of fluid pushed aside as the object moves some distance 'd' is \rho A d. Hence:

\frac 1 2 \rho A d v^2 = Fd

F = \frac 1 2 \rho A v^2