Find roots of polynomial $$ p(x) = x^4+4x^3+6x^2+5x $$

The roots of polynomial $ p(x) $ are:

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -2.3532\\[1 em]x_3 &= -0.8234+1.2028i\\[1 em]x_4 &= -0.8234-1.2028i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x }$ from $ x^4+4x^3+6x^2+5x $ and solve two separate equations:

$$ \begin{aligned} x^4+4x^3+6x^2+5x & = 0\\[1 em] \color{blue}{ x }\cdot ( x^3+4x^2+6x+5 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ x^3+4x^2+6x+5 & = 0 \end{aligned} $$

One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.

Step 2:

Polynomial $ x^3+4x^2+6x+5 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

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