# what is the result?what is the result?(x^2+1)-(x^2-4x+1)^2=? what has to be done to start easier?

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We have to find (x^2+1) - (x^2-4x+1)^2

(x^2+1) - (x^2-4x+1)^2

=> (x^2+1) - [(x^2 + 1)- 4x]^2

take the square of the terms within the brackets

=> (x^2+1) - [(x^2 + 1)^2 + 16x^2 - 8x*(x^2 + 1)]

open the brackets

=> x^2 + 1 - (x^2 + 1)^2 - 16x^2 + 8x^3 + 8x

=> x^2 + 1 - x^4 - 1 - 2x^2 - 16x^2 + 8x^3 + 8x

=> - x^4 + 8x^3 - 17x^2 + 8x

**The required result is - x^4 + 8x^3 - 17x^2 + 8x**

We recognize the difference of two squares and we'll apply the formula:

a^2 - b^2 = (a-b)(a+b)

We'll put a= x^2+1 and b = x^2-4x+1

(x^2+1)-(x^2-4x+1)^2 =( x^2+1- x^2+4x-1)( x^2+1+x^2-4x+1)

We'll combine like terms inside brackets:

(x^2+1)-(x^2-4x+1)^2 = (4x)( 2x^2+2-4x)

(x^2+1)-(x^2-4x+1)^2 = 8x(x^2 - 2x + 1)

We notice that inside brackets we have a perfect square:** **

**(x^2+1)-(x^2-4x+1)^2 = 8x(x - 1)^2**