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On solving the problem of mathematical factorization.
Solution: Let 2011= x2012 = y.

So m = 20 1 1? +20 1 1? ×20 12? +20 12? =x? +x? ×y? +y?

=x? +x? y? +y?

=(x? -2xy+y? )+(x? y? +2xy+ 1)- 1

=(x-y)? +(xy+ 1)? - 1

=(20 1 1-20 12)? +(xy+ 1)? - 1

= 1- 1+(20 1 1×20 12+ 1)?

=(20 1 1×20 12+ 1)?

∴m is a complete square.

(1) shows that this polynomial can be divisible by (x-2); (x-2) can be equal to 0; That is, x can be equal to 2.

(2)M is a multiple of the algebraic expression (x-k), and x can be equal to k.

Answer finished, hope to adopt, thank you.