sign of a polynomial

[glyphomouni]

Find the sign of the following polynomial (i.e whether is greater or less than zero)

 

[math]x^4+x^3(a+b)+x^2(a+ab+b)+x(a^2+b^2) +ab[/math],given that

 

[math]|a|<\sqrt{b}[/math] and [math]|b|<\sqrt{a}[/math]

 

 

Since [math] x^4 ,x^2[/math] are positive if we can show that (a+ab+b)>0,ab>0 and also [math]x^3(a+b)-x(a^2+b^2)>0[/math] are we done??

[shyvera]

Hint:

 

[math]\left(x^2+ax+b\right)\left(x^2+bx+a\right)[/math]