Simplify (x + 1)^3 - 2(x + 1)^2 (x - 1) - (x + 1)(x - 1)^2 + 2(x + 1)^3.

Step-by-step solution

1. 1

   Collect the repeated term first. The first and last terms are like terms in $(x+1)^3$:

   $(x+1)^3+2(x+1)^3=3(x+1)^3$

   so the expression is

   $3(x+1)^3-2(x+1)^2(x-1)-(x+1)(x-1)^2$

   Doing this before expanding removes a third of the work.

2. 2

   Pull out the common factor $(x+1)$. Every term contains at least one copy:

   $(x+1)\Big[3(x+1)^2-2(x+1)(x-1)-(x-1)^2\Big]$

3. 3

   Expand only the bracket. With $(x+1)^2=x^2+2x+1$, $(x+1)(x-1)=x^2-1$ and $(x-1)^2=x^2-2x+1$:

   $3\left(x^2+2x+1\right)-2\left(x^2-1\right)-\left(x^2-2x+1\right)=3x^2+6x+3-2x^2+2-x^2+2x-1$

   The $x^2$ terms give $3-2-1=0$, so the bracket is only linear:

   $8x+4$

4. 4

   Multiply back and factor.

   $(x+1)(8x+4)=8x^2+12x+4=4\left(2x^2+3x+1\right)=4(x+1)(2x+1)$

   The cubic terms cancelling is the real content here: a sum of degree-3 products has collapsed to degree $2$.

5. 5

   Verify at two values. At $x=0$: the original is $1-2(1)(-1)-1(1)+2=1+2-1+2=4$, and $4(1)(1)=4\;\checkmark$. At $x=2$: the original is $27-2(9)(1)-3(1)+54=27-18-3+54=60$, and $4(3)(5)=60\;\checkmark$.