<s>[INST] Find possible minimal polynomial for $L^3 + 2L^2 + L + 3 \cdot I_v$ Given:
$L: V \rightarrow V ; L^2 + I_v = 0$
Find:
possibilities for the minimal polynom of $L^3+2L^2+L+3\cdot I_v$
 [/INST] $x - 1$.  The one and only minimal polynomial of $L^3 + 2L^2 + L + 3I_V \;$ when $L^2 + I_V = 0$ is $x - 1$.
Observe that
$x^3 + 2x^2 + x + 3 = (x + 2)(x^2 + 1) + 1, \tag{1}$
which in turn implies, since $L^2 + I_V = 0$, 
$L^3 + 2L^2 + L + 3I_V = (L + 2I_V)(L^2 + I_V) + I_V = I_V; \tag{2}$
the one and only minimal polynomial of $I_V$ is $x - 1$.
Hope this helps.  Cheerio,
and as always,
Fiat Lux!!!
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