Is there any way to simplify it? I could only come up with $a^5+2a+1=-a^2+a+1$ but it still seems difficult to calculate

U can factorise

$x^5 = -x^2-x \implies x^5 + 2x + 1 = 1 + x - x^2 = y \implies (2x-1)^2 = 4y+5 $ Also $x^4 = (1+x-y)^2 \implies -1-x = 1+x^2+y^2+2x-2y-2xy$ $\implies y^2-2y-2xy+3x+2+1+x-y = 0 \implies 2x = \frac{y^2-3y+3}{2-y}$ using them we can get the new equation