### Instruction: Compute $a^2 + b^2 + c^2,$ given that $a,$ $b,$ and $c$ are the roots of \[2x^3 - x^2 + 4x + 10 = 0.\].### Response: To solve this problem, I need to use the fact that the roots of a cubic polynomial satisfy some relationships involving the coefficients.
Specifically, if $a,$ $b,$ and $c$ are the roots of $px^3 + qx^2 + rx + s = 0,$ then \[a + b + c = -\frac{q}{p},\] \[ab + bc + ca = \frac{r}{p},\] and \[abc = -\frac{s}{p}.\]
These are called Vieta's formulas, and they can be derived by expanding $(x-a)(x-b)(x-c) = 0$ and equating coefficients.
In this problem, I have $p = 2,$ $q = -1,$ $r = 4,$ and $s = 10,$ so I can apply Vieta's formulas to get \[a + b + c = \frac{1}{2},\] \[ab + bc + ca = 2,\] and \[abc = -5.\]
Now, I want to find $a^2 + b^2 + c^2,$ which is not directly given by any of the formulas.
However, I can use a clever trick to rewrite this expression in terms of the formulas I have.
I notice that \[(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca).\]
Therefore, \[a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca).\]
Now, I can plug in the values I obtained from Vieta's formulas and simplify.