<s>[INST] Getting a cubic equation from 3 equations Here is the problem:

Three numbers have a sum of $5$ and the sum of their squares is $29$.
If the product of the three numbers is $−10$, what are the three
numbers? Express your answer in simplest radical form.

I used Vieta's formulas to get $x^3-5x^2+12x+10=0$
What should I do after this?
 [/INST] As @WhatsUp noted in the comments, the equation should have been $x^3-5x^2-2x+10=0$, and so you can find by inspection or by the Rational Root Theorem that $x=5$ is a root. Divide and get $x^2-2=0$, and so your answers should be $x=5, \sqrt{2}, -\sqrt{2}$.
Edit: Another (essentially equivalent, but more detailed) method to get the coefficient of $-2$ for the linear term:
Assume the roots are $a, b, c$.
Square the first equation:
$(a+b+c=5)^2\Rightarrow (a^2+b^2+c^2+2(ab+ac+bc))=25$
We know that $a^2+b^2+c^2=29$ by the second equation, and so we get $2(ab+ac+bc)=-4\Rightarrow ab+ac+bc=-2$
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