<s>[INST] Factoring by grouping: $x^4 - y^4 -4x^2 + 4$ Please help me factor $x^4 - y^4 -4x^2 + 4$ by grouping terms.
Thank you.
 [/INST] $x^4 - y^4 -4x^2 + 4 $
$= (x^4 -4x^2 + 4) - y^4$
$= [(x^2)^2 -2(2x^2) + 2^2] - (y^2)^2$
The expression $(x^2)^2 -2(2x^2) + 2^2$ has the form: $a^2 -2ab + b^2 = (a - b)^2$
where $a=x^2$ and $b=2$
Thus, $(x^2)^2 -2(2x^2) + 2^2 = (x^2 - 2)^2$ 
Hence 
$[(x^2)^2 -2(2x^2) + 2^2] - (y^2)^2$ 
$= (x^2 - 2)^2 -(y^2)^2$ 
which is a difference of two squares i.e. has the form: $a^2-b^2 = (a+b)(a-b)$
where  $a= x^2-2$ and $ b = y^2$
Thus, $(x^2 - 2)^2 -(y^2)^2 = (x^2 -2 + y^2)(x^2 -2 - y^2)$
$\therefore $   $x^4 - y^4 -4x^2 + 4 = (x^2 -2 + y^2)(x^2 -2 - y^2)$ Answer
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