To find the equation whose roots are the squares of the roots of the given quadratic equation \(x^2 + x + 2 = 0\), follow these steps:

1. Find the roots of \(x^2 + x + 2 = 0\):

Use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 1\), \(b = 1\), and \(c = 2\).

Calculate the discriminant:
\[
b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot 2 = 1 - 8 = -7
\]
The discriminant is negative, so the roots are complex.

The roots are:
\[
x = \frac{-1 \pm \sqrt{-7}}{2} = \frac{-1 \pm i\sqrt{7}}{2}
\]
Let the roots be \( \alpha \) and \( \beta \):
\[
\alpha = \frac{-1 + i\sqrt{7}}{2}, \quad \beta = \frac{-1 - i\sqrt{7}}{2}
\]

2. Find the squares of the roots:

Compute \( \alpha^2 \) and \( \beta^2 \):
\[
\alpha^2 = \left(\frac{-1 + i\sqrt{7}}{2}\right)^2 = \frac{(-1 + i\sqrt{7})^2}{4}
\]
\[
(-1 + i\sqrt{7})^2 = 1 - 2i\sqrt{7} - 7 = -6 - 2i\sqrt{7}
\]
\[
\alpha^2 = \frac{-6 - 2i\sqrt{7}}{4} = -\frac{3}{2} - \frac{i\sqrt{7}}{2}
\]
Similarly,
\[
\beta^2 = \left(\frac{-1 - i\sqrt{7}}{2}\right)^2 = \frac{(-1 - i\sqrt{7})^2}{4} = -\frac{3}{2} + \frac{i\sqrt{7}}{2}
\]

3. Sum and product of \( \alpha^2 \) and \( \beta^2 \):

The sum \( \alpha^2 + \beta^2 \):
\[
\alpha^2 + \beta^2 = -\frac{3}{2} - \frac{i\sqrt{7}}{2} + -\frac{3}{2} + \frac{i\sqrt{7}}{2} = -3
\]

The product \( \alpha^2 \cdot \beta^2 \):
\[
\alpha \beta = \frac{-1 + i\sqrt{7}}{2} \cdot \frac{-1 - i\sqrt{7}}{2} = \frac{(-1)^2 - (i\sqrt{7})^2}{4} = \frac{1 + 7}{4} = 2
\]
\[
(\alpha \beta)^2 = 2^2 = 4
\]

4. Form the quadratic equation:

The quadratic equation with roots \( \alpha^2 \) and \( \beta^2 \) is given by:
\[
x^2 - (\text{Sum of the roots}) \cdot x + (\text{Product of the roots}) = 0
\]
\[
x^2 - (-3) \cdot x + 4 = 0
\]
\[
x^2 + 3x + 4 = 0
\]

So, the correct equation is:

A. \(x^2 + 3x + 4 = 0\)