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To find the sum of the polynomials \(3x^2 + 2x + 1\) and \(2x^2 - x + 4\), we will add them together by combining like terms.
1. **Combine the \(x^2\) terms:**
\[
3x^2 + 2x^2 = 5x^2
\]
2. **Combine the \(x\) terms:**
\[
2x - x = x
\]
3. **Combine the constant terms:**
\[
1 + 4 = 5
\]
Now, combine all these results:
\[
5x^2 + x + 5
\]
Thus, the sum of the polynomials is \(5x^2 + x + 5\).
**The correct answer is A: \(5x^2 + x + 5\)**.