Question

multiply and simplify: (x^2+2x+2)(x^2-2x+2)

Answer 1

method or answer?

Answer 2

answer is
\[x^4+4\]

Answer 3

method

Answer 4

what is more interesting that multiplying this out doing 9 multiplications is to turn
\[x^4+4\] into
\[(x^2+2x+2)(x^2-2x+2)\]

Answer 5

you do it as follows. add and subtract
\[4x^2\] and write
\[x^4+4x^2+4-4x^2\] which is the difference of two squares

Answer 6

now you have
\[(x^2+2)^2-4x^2\] and the difference of two squares factors so you get
\[(x^2+2-2x)(x^2+2+2x)\] which is what you started with

Answer 7

this is sometimes called "completing the square" or the "sophie germain" trick

Answer 8

otherwise you just have to multiply out and it is a pain.
\[(x^2+2x+2)(x^2-2x+2)\]
\[x^4-2x^3+2x^2+2x^3-4x^2+4x+2x^2-4x+4\] and collect terms to get your answer