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1 solution
Solution 1:
x 4 + 2 x 3 + 3 x 2 + 2 x + 1 = ( x 2 + x + 1 ) 2 = 0
⇒ x 2 + x + 1 = 0
⇒ ( x + 1 ) 2 − x = 0
⇒ ( x + 1 ) 2 = x
Since ( x + 1 ) 2 ≥ 0 , x has to be a non-negativ number, but it contradict that x 2 + x + 1 = 0 . So the equation hasn't got any solution.
Solution 2:
( x 4 + 2 x 3 + x 2 ) + x 2 + ( x 2 + 2 x + 1 ) = 0
⇒ x 2 ( x + 1 ) 2 + x 2 + ( x + 1 ) 2 = 0
On the left side there are only perfect squares, so the left side is non-negativ. If the sum of three non-negativ numbers is 0 , then all of them is 0. But x 2 and ( x + 1 ) 2 can't be 0 at the same time, so the equation hasn't got any solution.