factorise ^{x4 +x2 +1 }

Let x^{2}= p

Then x^{4}= p^{2}

x^{4}+ x^{2}+ 1

= p^{2}+ p + 1

= p^{2}+ 1 + p

= (p)^{2}+ (1)^{2}+ 2*p*1 - p

= (p)^{2}+ (1)^{2}+ 2p - p

= (p + 1)^{2}- p

= (p + 1)^{2}- (√p)^{2}

= (p + 1 + √p) (p + 1 - √p)

Substituting the value of p,

( x^{2}+ 1 + √x^{2}) (x^{2}+ 1 - √x^{2})

= (x^{2}+ 1 +x) (x^{2}+ 1 - x)