All polynomials #P(x)# can be rewritten in terms of three other polynomials #Q(x)# (quotient), #D(x)# (divisor), and #R(x)# (remainder):
#(P(x))/(D(x))=Q(x)+(R(x))/(D(x))#.
This can be rewritten to #P(x)=Q(x)D(x)+R(x)#. Now, suppose that there is a constant #k# such that #D(k)=0#. Then, it follows that #P(k)=R(k)#.
This is the Remainder Theorem: if a polynomial #P(x)# is divided by #D(x)# with #D(k)=0#, then the remainder #R(x)# satisfies #R(k)=P(k)#.
If #x+2# is a factor of #3x^3-2x^2-6x-2#, then the remainder of #3x^3-2x^2-6x-2# divided by #x+2# must be #0#.
According to the remainder theorem, the remainder #R# (which is a constant number in this case as the divisor is linear) satisfies #R=P(k)#, where #k# is the solution to #x+2=0#, i.e. #-2#. Thus, #R=3(-2)^3-2(-2)^2-6(-2)-2=-22#. Since #R!=0#, #x+2# is not a factor of #3x^3-2x^2-6x-2#.
