Question

\(x^4+3x^3+4x^2+3x+1=0\)

Answer 1

\(\Leftrightarrow\left(x^4+x^3\right)+\left(2x^3+2x^2\right)+\left(2x^2+2x\right)+\left(x+1\right)\)\(=0\)

\(\Leftrightarrow x^3\left(x+1\right)+2x^2\left(x+1\right)+2x\left(x+1\right)+\left(x+1\right)=0\)

\(\Leftrightarrow\left(x+1\right)\left(x^3+2x^2+2x+1\right)=0\)

\(\Leftrightarrow\left(x+1\right)\left(\left(x^3+x^2\right)+\left(x^2+x\right)+\left(x+1\right)\right)=0\)

\(\Leftrightarrow\left(x+1\right)^2\left(x^2+x+1\right)=0\)

\(Do\)\(x^2+x+1=\left(x+\frac{1}{2}\right)^2+\frac{3}{4}\ge0\)

\(\Rightarrow\left(x+1\right)^2=0\)

\(\Leftrightarrow x+1=0\)

\(\Leftrightarrow x=-1\)