Evaluate $(2009)^2 - (2008)(2010)$.

$$\small{\text{Evaluate $(2009)^2 - (2008)(2010). $}}$$

I.

$$\small{\text{$ \begin{array}{rcl} (2009)^2 - (2008)(2010) \\ &=& 2009^2-(2009-1)(2009+1)\\ &=& 2009^2-(2009^2-1)\\ &=& 2009^2 - 2009^2 + 1 \\ &=& 1 \end{array} $}}$$

II.

$$\small{\text{$ \begin{array}{rcl} (2009)^2 - (2008)(2010) \\ &=& (2008+1)^2 - 2008\cdot 2010 \\ &=& 2008^2 + 2\cdot 2008 + 1 - 2008\cdot 2010\\ &=& 2008^2 - 2008 (-2+2010) + 1 \\ &=& 2008^2 - 2008^2 + 1 \\ &=& 1 \end{array} $}}$$

III.

$$\small{\text{$ \begin{array}{rcl} (2009)^2 - (2008)(2010) \\ &=& (2010-1)^2 - 2008\cdot 2010 \\ &=& 2010^2 - 2\cdot 2010 + 1 - 2008\cdot 2010\\ &=& 2010^2 - 2010 (2+2008) + 1 \\ &=& 2010^2 - 2010^2 + 1 \\ &=& 1 \end{array} $}}$$