1. **State the problem:** Given the system of equations: $$a+b+c=5$$ $$ab+bc+ca=10$$ Find possible values or relationships for $a$, $b$, and $c$. 2. **Recall the formulas:** For three variables $a$, $b$, and $c$, the expressions $a+b+c$, $ab+bc+ca$, and $abc$ are the elementary symmetric sums related to the roots of a cubic polynomial: $$x^3 - (a+b+c)x^2 + (ab+bc+ca)x - abc = 0$$ 3. **Use given values:** Substitute the known sums: $$x^3 - 5x^2 + 10x - abc = 0$$ 4. **Interpretation:** Without the value of $abc$, we cannot find exact roots, but we know the polynomial has roots $a$, $b$, and $c$. 5. **Additional insight:** The discriminant or further conditions would be needed to find specific values. 6. **Summary:** The problem provides two symmetric sums; the third symmetric sum $abc$ is unknown, so the roots satisfy: $$x^3 - 5x^2 + 10x - abc = 0$$ This is the polynomial with roots $a$, $b$, and $c$. **Final answer:** The values $a$, $b$, and $c$ are roots of the cubic equation $$x^3 - 5x^2 + 10x - abc = 0$$ where $abc$ is unknown from the given data.