Ta có : x4+3x3+4x2+3x+1=0
⇔ ( x4 + x3 ) + ( 2x3 + 2x2 ) + ( 2x2 + 2x ) + ( x + 1 ) = 0
⇔ x3 ( x + 1 ) + 2x2 ( x + 1 ) + 2x ( x+1 ) + ( x + 1 ) =0
⇔ ( x + 1 ) ( x3 + 2x2 + 2x + 1 ) = 0
⇔ ( x + 1 ) [ ( x3 + 1 ) + ( 2x2 + 2x ) ] = 0
⇔ ( x + 1 ) [ (x + 1 ) ( x2 - x +1 ) + 2x ( x + 1 ) ] =0
⇔ ( x +1 ) ( x + 1 ) ( x2 + x +1 ) =0
⇒ \(\left[{}\begin{matrix}x+1=0\\x^{2^{ }}+x+1=0\end{matrix}\right.\)<=> \(\left[{}\begin{matrix}x=-1\\\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}=0\left(VoLy\right)\end{matrix}\right.\)
Vậy x = -1
x4+3x3+4x2+3x+1=0
⇔(x4+2x3+x2)+(x3+2x2+1)+(x2+2x+1)=0
⇔x2(x2+2x+1)+x(x2+2x+1)+(x2+2x+1)=0
⇔x2(x+1)2+x(x+1)2+(x+1)2=0
⇔(x+1)2(x2+x+1)=0
Vì x2+x+1=x2+x+\(\dfrac{1}{4}\)+\(\dfrac{3}{4}\)=(x+\(\dfrac{1}{2}\))2+\(\dfrac{3}{4}\)>0 nên phương trình đã cho tương đương:
(x+1)2=0 ⇔(x+1)(x+1)=0 ⇔x=-1.