http://www.youtube.com/watch?v=a7yenMwLzuU
this video is a pretty good explanation of factoring by decomposition method.
here's how i did it, and i hope i can convey what i mean.
question to solve is:
8k^2-10k-3
the way my teacher tells us is to look at the parts of the equation, 8k^2, -10k, -3.
you have to first multiply the first and last terms of the equation 8(-3)=-24
then find out what two terms multiply to -24, and add to -10.
my first guess was -12 and 2, (-12 x 2=-24 & -12+2=-10) but i couldn't factor it out right (but i later got the answer even though factoring is one of my weakest section in math) so i also guessed -6 & +/-4...which i somehow figured out to = 24 and -10 but 6 & 4 are actually deceptive values. but i factored it out anyways and even though it kinda looks like your solution it's not the right one.
here's the wrong one first using -6 & -4
8x^2-10x-3
=8x^2-6x-4x-3
=(8x^2-6x)-(4x-3)
=2x(4x-3)-(4x-3)------>(4x-3) is repeated so there's no need to repeat it in the next step. you just keep one version of it because it cancels out.
=(2x-1)(4x-3)------> -1 comes from the invisible value of 1 that is infront of the bracket: -1(4x-3). note that -1 multiplied into 4x-3 actually becomes 4x+1.
here's the right way using the right values of -12 & 2 knowing that (-12 x 2=-24 & -12+2=-10)
8x^2-10x-3
=8x^2-12x+2x-3
=(8x^2-12x)+(2x-3)
=4x(2x-3)+(2x-3)------>(2x-3) is repeated so there's no need to repeat it in the next step. you just keep one version of it because it cancels out.
=(4x+1)(2x-3)------> +1 comes from the invisible value of 1 that is infront of the bracket: +1(2x-3). note that 1multiplied into 2x-3 just gives you 2x-3.
