`A=x^{2010}-2012x^{2009}+2012x^2008-2012x^{2007}+...-2012x^2+2012x-1`
`x=2011`
`⇒x=2012-1`
`⇒x+1=2012`
`⇒A=x^{2010}-(x+1)x^{2009}+(x+1)x^{2008}-(x+1)x^{2007}+...-(x+1)x^2+(x+1)x-1`
`⇒A=x^2010-x^2010-x^2009+x^2009+x^2008-x^2008-x^2008+...-x^3-x^2+x^2+x-1`
`⇒A=x-1`
`⇒A=2011-1`
`⇒A=2010`