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A second-degree polynomial, often known as a quadratic polynomial, takes the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). This type of polynomial is characterized by having its highest exponent as 2, which indicates it will graph as a parabola on a coordinate plane.

When solving an equation involving a second-degree polynomial, the goal is often to factor it into the product of two linear factors. The reason for this is straightforward: it can simplify many tasks such as finding the roots of the polynomial, integrating it, or analyzing its graph. Factoring second-degree polynomials can be thought of as reversing the process of expanding two binomials.

Linear factors refer to expressions of the first degree, meaning they have no exponents higher than 1. A linear factor is in the form \(dx + e\), where \(d\) and \(e\) are also constants. The importance of representing a polynomial as the product of linear factors lies in their simplicity: linear factors are easy to understand and handle mathematically.

The act of breaking down a polynomial into linear factors is essentially revealing the x-intercepts of its graph on a coordinate plane. The second-degree polynomial from our exercise can be factored into \((3x - 2)(x - 1)\). These factors correspond to the points where the graph will cross the x-axis, which are vital in understanding the algebraic and graphical properties of the polynomial.

The product-sum method is a strategy used to factor second-degree polynomials-in cases where it's applicable. It involves finding two numbers that multiply together to get the product (the coefficient of \(x^2\) times the constant term) and at the same time add up to get the sum (the coefficient of \(x\)).

In our exercise, using the product-sum method, we need to find two numbers that multiply to 6 (because \(3 * 2 = 6\)) and add to -5. These two numbers are -3 and -2. Once we have these numbers, we can write our original polynomial as a product of the binomials \((3x - 2)(x - 1)\). This method greatly simplifies the process of factoring and is a fundamental skill in algebra.