Answer
$x=\{-3, \pm i, \pm1 \}$
Work Step by Step
Using factoring by grouping, the given equation, $
x^5+3x^4=x+3
$, is equivalent to
\begin{array}{l}\require{cancel}
x^5+3x^4-x-3=0
\\\\
(x^5+3x^4)-(x+3)=0
\\\\
x^4(x+3)-(x+3)=0
\\\\
(x+3)(x^4-1)=0
\\\\
(x+3)(x^2+1)(x^2-1)=0
.\end{array}
Equating each factor to $0$, then,
\begin{array}{l}\require{cancel}
x+3=0
\\\\
x=-3
,\\\\\text{OR}\\\\
x^2+1=0
\\\\
x^2=-1
\\\\
x=\pm\sqrt{-1}
\\\\
x=\pm i
,\\\\\text{OR}\\\\
x^2-1=0
\\\\
x^2=1
\\\\
x=\pm\sqrt{1}
\\\\
x=\pm1
.\end{array}
Hence, $
x=\{-3, \pm i, \pm1 \}
.$
