First, take out a constant number. All the coefficients have a common factor #4#:
#color(white)=8x^5+28x^4+12x^3#
#=color(red)4*2x^5+color(red)4*7x^4+color(red)4*3x^3#
#=color(red)4(2x^5+7x^4+3x^3)#
Next, take out the common #x^3# factor:
#=color(red)4(color(blue)(x^3)*2x^2+color(blue)(x^3)*7x+color(blue)(x^3)*3)#
#=color(red)4color(blue)(x^3)(2x^2+7x+3)#
Now, to factor the quadratic, find two numbers that multiply to #2*3#, or #6#, and add up to #7#.
After some experimentation, you will find that these two numbers are #1# and #6#. Split up the #x# terms into these numbers, then factor the first two terms and last two terms:
#=color(red)4color(blue)(x^3)(2x^2+6x+x+3)#
#=color(red)4color(blue)(x^3)(color(green)(2x)(x+3)+x+3)#
#=color(red)4color(blue)(x^3)(color(green)(2x)(x+3)+color(purple)1(x+3))#
#=color(red)4color(blue)(x^3)((color(green)(2x)+color(purple)1)(x+3))#
#=color(red)4color(blue)(x^3)(color(green)(2x)+color(purple)1)(x+3)#
That's the completely factored polynomial. Hope this helped!
