x² + x(1+y) + y² + y +1 = [ (x+1/2(1+y))]² - [1/2(1+y)]² + y² + y +1
= [ (x+1/2(1+y))]² -1/4(1+2y+y²) + y² + y +1
= [ (x+1/2(1+y))]² -1/4 - 1/2y-1/4y² + y² + y +1
= [ (x+1/2(1+y))]² +3/4y² + 1/2y + 3/4
= [ (x+1/2(1+y))]² + 3/4 ( y² + 2/3y + 1 )
= [ (x+1/2(1+y))]² + 3/4 [( y+1/3)² - 1/9 +1 ) ]
= [ (x+1/2(1+y))]² + 3/4[( y+1/3)² + 8/9 ] ≥ 0
L'idée c'est d'utilisé 2 fois la forme canonique .
Bon courage !
