Answer
See answer below
Work Step by Step
We are given $S=\{x^4+x^2+1,x^2+x+1,x+1,x^4+2x+3\}$ in $P^4$
Assume $x^4+x^2+1=v_1\\
x^2+x+1=v_2 \\
x+1=v_3\\
x^4+2x+3=v_4$
Since $dim P_4=5$, set $S$ does not span $P_4$
We can notice that $v_1-v_2+3v_3=x^4+2x+3=v_4$
Hence $S$ is linearly independent in $P_4$
