2(x+1)(x-3)^{2 }-3(x+1)^{2 }(x-3)

First you have to factor out common terms with the smallest exponents, in this case (x+1) appears with exponents 1 and 2, and (x-3) appears also with exponents 2 and 1, so we factor each of them with exponent 1

(x+1)(x-3) [ 2(x-3) - 3(x+1) ] =

*(verify this by multiplying (x+1)(x-3) by [...] and using the distributive property)*

= (x+1)(x-3) [ 2x-6 -3x -3 ] (applying distributive property twice inside the [ ])

= (x+1)(x-3) (-x - 9) (grouping inside of the [ ])

and that's that!

Taking the minus sign out from the last factor, you can also write the answer as

- (x+1)(x-3)(x+9) (notice the minus sign in front and make sure you understand how it was done)

Any questions?