1. The problem asks for the values of $k$ such that the quadratic equation $2x^2 + kx + 2 = 0$ has no real roots. 2. Recall that a quadratic equation $ax^2 + bx + c = 0$ has no real roots if its discriminant is less than zero. The discriminant $\Delta$ is given by: $$\Delta = b^2 - 4ac$$ 3. For our equation, $a = 2$, $b = k$, and $c = 2$. Substitute these into the discriminant formula: $$\Delta = k^2 - 4 \times 2 \times 2 = k^2 - 16$$ 4. For no real roots, we require: $$k^2 - 16 < 0$$ 5. Solve the inequality: $$k^2 < 16$$ 6. Taking square roots: $$-4 < k < 4$$ 7. Therefore, the values of $k$ for which the quadratic has no real roots are those strictly between $-4$ and $4$. Final answer: $-4 < k < 4$