We just see that the given polynomial is $2x^5-(x+1)^5$ and we are done.

If the computer is still needed, than the following lines investigate the factorization and the roots in sage:

```
sage: K.<z> = CyclotomicField( 5 )
sage: RK.<X> = K[]
sage: F.<a> = K.extension( X^5-2 )
sage: RF.<x> = F[]
sage: P = x^5 - 5*x^4 - 10*x^3 -10*x^2 - 5*x - 1
sage: for f in P.factor(): print f
(x + (z^3 + z^2 + z + 1)*a^4 - z^3*a^3 - z^2*a^2 - z*a - 1, 1)
(x - z^2*a^4 + (z^3 + z^2 + z + 1)*a^3 - z*a^2 - z^3*a - 1, 1)
(x - z*a^4 - z^2*a^3 - z^3*a^2 + (z^3 + z^2 + z + 1)*a - 1, 1)
(x - z^3*a^4 - z*a^3 + (z^3 + z^2 + z + 1)*a^2 - z^2*a - 1, 1)
(x - a^4 - a^3 - a^2 - a - 1, 1)
sage: P.roots( multiplicities=False)
[(-z^3 - z^2 - z - 1)*a^4 + z^3*a^3 + z^2*a^2 + z*a + 1,
z^2*a^4 + (-z^3 - z^2 - z - 1)*a^3 + z*a^2 + z^3*a + 1,
z*a^4 + z^2*a^3 + z^3*a^2 + (-z^3 - z^2 - z - 1)*a + 1,
z^3*a^4 + z*a^3 + (-z^3 - z^2 - z - 1)*a^2 + z^2*a + 1,
a^4 + a^3 + a^2 + a + 1]
```

In the above, $z$ is $\zeta=\zeta_5$, a primitive root of order $5$ of one. And $a$ is $2^{1/5}$.

The five roots are found manually by starting with the equation $2x^5=(x+1)^5$, dividing by $x^5$, so we get equivalently $\left(1+\frac 1x\right)^5=2$. So $\left(1+\frac 1x\right)^5=a\zeta^k$ for $k$ among $0,1,2,3,4$. This gives immediately:
$$ x = -\frac 1{1-a\zeta^k}
= \frac {1-(a\zeta^k)^5}{1-a\zeta^k}
= 1
+a\zeta^k
+a^2\zeta^{2k}
+a^3\zeta^{3k}
+a^4\zeta^{4k}\ .
$$
One of the roots is thus $1+a+a^2+a^3+a^4$. For $k=0$.
The others are obtained from it by replacing $a$ with a (conjugate) root of $X^5-2$.

Note:

If the coin is not immediately falling while typing the polynomial, then asking

```
sage: R.<x> = QQ[]
sage: P = x^5 - 5*x^4 - 10*x^3 -10*x^2 - 5*x - 1
sage: P.galois_group()
```

may help. I've got an error mess...

verbose 0 (2072: permgroup_named.py, cardinality) Warning: TransitiveGroups requires the GAP database package. Please install it with `sage -i database_gap`

.

But instead of making it work, the following was enough for me:

```
dan ~$ gp
Reading GPRC: /etc/gprc ...Done.
PARI/GP is free software, covered by the GNU General Public License, and comes
WITHOUT ANY WARRANTY WHATSOEVER.
? P = 2*x^5 - (x+1)^5
%1 = x^5 - 5*x^4 - 10*x^3 - 10*x^2 - 5*x - 1
? polgalois( P )
%2 = [20, -1, 1, "F(5) = 5:4"]
? ?polgalois
polgalois(T): Galois group of the polynomial T (see manual for group coding).
Return [n, s, k, name] where n is the group order, s the signature, k the index
and name is the GAP4 name of the transitive group.
```

The order is making the group solvable already.