Multiply both sides by LCD: $$x(x + 2)(\frac{2}{x} + \frac{3}{x + 2}) = \frac{4}{x}x(x + 2)$$ $$2(x + 2) + 3x = 4(x + 2)$$ $$2x + 4 + 3x = 4x + 8$$ $$5x - 4x = 8 - 4$$ $$x = 4$$

The given equation is not a linear equation; the equation is a quadratic equation. Applying the quadratic formula, we can solve it.

The general form of a quadratic equation is \(ax^{2} + bx + c = 0\). Here, \(a, b, c\) are constants and \(a \neq 0\).

Quadratic equations can be solved by either factoring the equation or using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$

Substituting \(a = 1\), \(b = 13\), & \(c = 30\) in the quadratic formula, we get $$x = \frac{-13 \pm \sqrt{13^{2} - 4\cdot 1\cdot 30}}{2\cdot 1}$$ $$x = \frac{-13 \pm \sqrt{49}}{2}$$ $$x = \frac{-13 \pm 7}{2}$$ $$x = -10 \qquad x = -3$$

This problem can also be solved using the factoring approach. $$x^{2} + 13x + 30 = 0$$ $$x^{2} + 10x + 3x + 30 = 0$$ $$x(x + 10) + 3(x + 10) = 0$$ $$(x + 10)(x + 3) = 0$$ $$x = -10 \qquad x = -3$$