Simplify the Following Expression
\[ (x^2-x+1)^2-(x^2+x+1)^2 \]
Solution:
Using identity:
\[ a^2-b^2=(a-b)(a+b) \]
\[ = \left[(x^2-x+1)-(x^2+x+1)\right] \left[(x^2-x+1)+(x^2+x+1)\right] \]
\[ = (-2x)(2x^2+2) \]
\[ = -4x(x^2+1) \]
\[ (x^2-x+1)^2-(x^2+x+1)^2 \]
Using identity:
\[ a^2-b^2=(a-b)(a+b) \]
\[ = \left[(x^2-x+1)-(x^2+x+1)\right] \left[(x^2-x+1)+(x^2+x+1)\right] \]
\[ = (-2x)(2x^2+2) \]
\[ = -4x(x^2+1) \]