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How can I find the roots of a quintic equation. However my original question is what is the sum of the roots of $x^5+3x^2+7=0$.

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    the sum of the roots is zero. – 2017-01-17
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    For finding the sum of the roots of general polynomials [Vieta's formulas](https://en.wikipedia.org/wiki/Vieta's_formulas) are useful – 2017-01-17

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The roots do not need to be found explicitly.

Hint:

Suppose the polynomial factors as $f(x) = (x-a_1)(x-a_2) \cdots (x-a_5)$. Now, if you were to expand this, what is the coefficient on the $x^4$ term?

Notice that the constant term also has an interesting piece of information. In general, the coefficients on these terms are elementary symmetric polynomials evaluated at the roots.

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    Not that it matters here, but that coefficient would be the sum of the roots *counting multiplicities* ;-) – 2017-01-17
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    Indeed, that's a good point to make @dxiv. Even without finding the roots explicitly, one can check whether $f$ has repeated roots by computing $\gcd(f, f')$. – 2017-01-17