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The quadratic formula is a powerful tool for solving quadratic equations, which are polynomial equations of the second degree, meaning they have the general form of \(ax^2 + bx + c = 0\). When faced with a quadratic equation, the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] can be used to find the values of \(x\) that make the equation true.

This formula is derived from the process of completing the square and is applicable to any quadratic equation. To utilize the quadratic formula, you simply identify the coefficients \(a\), \(b\), and \(c\) from the equation and substitute them into the formula. It guarantees to provide the solutions for \(x\) whether they are real or complex numbers. In the example given, \(a = 3\), \(b = 6\), and \(c = -2\). Plugging these values into the formula yields the roots \( x= -2/3\) and \(1\), thus solving the quadratic equation.

At its most basic, a polynomial equation is an algebraic expression composed of variables, coefficients, and exponents, set equal to zero. The degree of the polynomial is determined by the highest power of the variable \(x\). Polynomial equations are a major subject in algebra and are the basis of many mathematical problems and applications

Quadratic equations are a subset of polynomial equations, where the variable \(x\) has a degree of two, that is \(x^2\). The general form of a quadratic polynomial equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). The given exercise is an example of a polynomial equation, which has been simplified to a quadratic equation, preparing it to be solved easily.

Factoring is another method used to solve quadratic equations that can be broken down into a product of simpler polynomials. The goal of factoring is to find two binomials that, when multiplied together, give you the original quadratic equation. This method is particularly useful when the quadratic equation can be easily decomposed into factors.

For the equation in the exercise, we could attempt to factor the simplified form \(3x^2 +6x -2 = 0\), to find solutions for \(x\). However, not all quadratics are factorable using rational numbers. If the quadratic cannot be factored or if factoring is difficult, other methods such as completing the square or using the quadratic formula are more appropriate. In our example, factoring is not straightforward, so the quadratic formula is the chosen method to solve for \(x\).