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It makes sense: it tried to solve symbolically, and it didn't get very far (it isolated the square, and took ± the square root).

You don't want to solve symbolically but numerically. You want to find the two real zeros of x^2 - 5 - sin(x), and from the picture you (roughly) know intervals where these are located, so you can use find_root:

sage: find_root(x^2 - 5 - sin(x), -5, 0)
-2.025211637444818
sage: find_root(x^2 - 5 - sin(x), 0, 5)
2.3846766601465696