Question

Sa se arate ca
$( {x}^{2} + 4x + 5)( {x}^{2} + 2x + 2) \geqslant 1$
Oricat ar fi x număr real

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Answer 1

(x²+4x+5)(x²+2x+2)≥1  ?

x²+4x+5=(x²+4x+4)+1=(x+2)²+1≥1

x²+2x+2=(x²+2x+1)+1=(x+1)²+1≥1

=> [(x+2)²+1] [(x+1)²+1]≥1

=> (x²+4x+5)(x²+2x+2)≥1