This is wrong unless there is some condition given on $e$, just consider $e\rightarrow \infty$.

thanks for the comment, Imomath had a typo, looking in the Abel site, i found the missing term +e^2 , in the inequality, now it is corrected

If $b+c+d+e=0$, we're done. Otherwise, as the required inequality is homogeneous, we may assume $b+c+d+e = 1$. Then $b^2+c^2+d^2+e^2 \ge 1/4$ by Cauchy-Schwarz, and it is enough to prove $a^2-a+1/4 = (a-1/2)^2 \ge 0$, which is true.