Correct Answer: Option B

Explanation

For the quadratic equation \(2x^2 + kx + 5 = 0\), the sum of roots \(\alpha + \beta = -\frac{k}{2}\) and the product \(\alpha \beta = \frac{5}{2}\).

Given \(\alpha^2 + \beta^2 = -1\), use the identity:
\(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\)

Substitute the values:
\(\left(-\frac{k}{2}\right)^2 - 2 \cdot \frac{5}{2} = -1\)
\(\frac{k^2}{4}\) - 5 = -1,
\(\frac{k^2}{4}\) = 4,
\(k^2\) = 16,
k = \(\pm\) 4.

The values of \(k\) are \(\pm 4\).

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