Inconsistency in Galois Theory?

I am a high school student who, after some tinkering, came across the Abel–Ruffini theorem. I then learned about its explanation through Galois theory, particularly the result that a polynomial is solvable by radicals if and only if its Galois group is solvable.

This leads to a confusion. The polynomial $x^5 +x +1$ is said to have Galois group (S5), which is not a solvable group. Therefore, I expected that its roots could not be expressed using radicals.

However, I noticed that

$$x^5+x+1=(x^2+x+1)(x^3-x^2+1).$$

The quadratic factor has no real roots, but the cubic factor certainly has a real root, and since cubic equations can be solved using Cardano's formula, this root can be expressed in radicals.

A similar situation occurs with $x^5+x-1$ .

I can also verify these factorizations and numerical roots using computational tools such as WolframAlpha. My confusion is this: if these quintic polynomials have factors that are solvable by radicals (and therefore have some roots expressible in radicals), how can their Galois groups be the unsolvable group ($S5$)?

Where exactly is my understanding of the connection between Galois groups and solvability going wrong?