Perhaps this

n = 10; v0 = 20;
y = Sum[(v[i] - v0/Xs*(Xs - x[i])^3)^2, {i, 0, n}];
y1 = Expand[D[y, Xs] Xs^3];(*Remove Xs from denominators*)
Simplify[Reduce[y1 == 0, Xs]]

All of the details of determining the sixth power polynomial in Xs has been done for you. With your values for x[i] and v[j] perhaps you can determine which roots are Real.

Or if the Root notation is confusing then use your values of x[i] and v[j] to determine the real roots of

440 Xs^6 -
Xs^5 Sum[180 x[i], {i, 0, 10}] -
Xs^4 Sum[2 v[i] - 300 x[i]^2, {i, 0, 10}] +
Xs^3 Sum[3 x[i] v[i] - 200 x[i]^3, {i, 0, 10}] -
Xs Sum[x[i]^3 v[i] - 60 x[i]^5, {i, 0, 10}] -
Sum[20 x[i]^6, {i, 0, 10}]

which is just a manual reformatting of the result from the code above.

Plot might even help understanding how many real roots there are or Reduce might do that if you define all your x[i] and v[j] values first.

(1) If you do not provide full code to replicate the example then it is impossible to really say much, short of retyping from an image (which I for one will not do).

(2) You will have a better chance if you make the upper bound an explicit number, 10, in this case, rather than encode it in the equations. Solve sees the equations but Sum does not, and Solve needs Sum to fully evaluate (to something explicit rather than a Sum).

(3) It is not a good idea to use N as a variable name, since it is also a heavily used function in Mathematica.