To solve this inequality, we first need to get rid of the fraction by multiplying both sides by 3:

[ 3 \left(\frac{x^2}{2}\right) \geqslant 3 \left(\frac{2x + 2}{3}\right) ]

This simplifies to:

[ \frac{3x^2}{2} \geqslant 2x + 2 ]

Now, we can simplify further by multiplying through by 2 to get rid of the fraction:

[ 3x^2 \geqslant 4x + 4 ]

Rearranging the terms:

[ 3x^2 - 4x - 4 \geqslant 0 ]

Now, we have a quadratic inequality. We can solve this inequality by finding the roots of the quadratic equation:

[ 3x^2 - 4x - 4 = 0 ]

The roots of this equation can be found using the quadratic formula:

[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(-4)}}{2(3)} ]
[ x = \frac{4 \pm \sqrt{16 + 48}}{6} ]
[ x = \frac{4 \pm \sqrt{64}}{6} ]
[ x = \frac{4 \pm 8}{6} ]

So, the roots are:

[ x = \frac{4 + 8}{6} \quad \text{and} \quad x = \frac{4 - 8}{6} ]
[ x = \frac{12}{6} \quad \text{and} \quad x = \frac{-4}{6} ]
[ x = 2 \quad \text{and} \quad x = -\frac{2}{3} ]

The solution to the inequality (3x^2 - 4x - 4 \geqslant 0) is (x \leqslant -\frac{2}{3}) or (x \geqslant 2).