Yes, since there is a power of "u" in each term, you can factor out u: \(\displaystyle u^2- 4u= u(u- 4)\). As JeffM said (a bit sharply) you should know that well before taking Calculus or "Pre-Calculus".
However, you should also be aware that this does not help with finding the limit, as u goes to 3.
Since (u^2- 4u)/(3u+ 5) (note the parentheses which positioned differently than in your fraction) is a "rational" function, you should know that such a function is continuous where ever it is defined. And the only way a fraction is not defined is where it its denominator is 0. At x= 3, the denominator of this fraction is 3(3)+ 5= 14, NOT 0. When u= 3, the numerator is \(\displaystyle 3^2- 4(3)= 9- 12= -3\). The limit, as u goes to 3, of (u^2- 4u)/(3u+ 5) is -3/14. No factoring is required nor does it help.