The volume of the solid formed by rotating the region about the x-axis is given by the integral:

\(\text{Vol} = \pi \int_{1}^{2} y^2 \, dx\)

Substitute \(y = x^2 + \frac{2}{x}\) into the equation:

\(\text{Vol} = \pi \int_{1}^{2} \left(x^2 + \frac{2}{x}\right)^2 \, dx\)

Expand the integrand:

\(\left(x^2 + \frac{2}{x}\right)^2 = x^4 + 4 + \frac{4x}{x^2}\)

\(= x^4 + 4x + \frac{4}{x^2}\)

Integrate term by term:

\(\int x^4 \, dx = \frac{x^5}{5}\)

\(\int 4x \, dx = 2x^2\)

\(\int \frac{4}{x^2} \, dx = -\frac{4}{x}\)

Thus, the integral becomes:

\(\text{Vol} = \pi \left[ \frac{x^5}{5} - \frac{4}{x} + 2x^2 \right]_{1}^{2}\)

Evaluate the definite integral:

\(\text{Vol} = \pi \left( \left[ \frac{32}{5} - 2 + 8 \right] - \left[ \frac{1}{5} - 4 + 2 \right] \right)\)

\(= \pi \left( \frac{71}{5} \right)\)

Therefore, the volume is \(\frac{71\pi}{5}\) or approximately 44.6.