First step: Realize that #x# needs to be multiplied by #x^2# in order to be the #x^3# in #x^3-10x^2+30x+3#. Then, multiply the #x^2# throughout #x-5# to get #x^3-5x^2# and SUBTRACT it (remember, the signs will change) from #x^3-10x^2+30x+3#.
You are left with #-5x^2+30x+3#, but only the first term is important for now. Like you did with the #x^3# in the first step, realize that you will need to add a #-5x# on top so that the #x(-5x)=-5x^2#. Continue multiplying and subtracting until you get to 28, which is the remainder.
All together, the answer would be written as:
#color(blue)(x^2-5x+5+28/(x-5)#
