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In algebra, a **common factor** is a number or a variable that divides each term in a polynomial evenly, without leaving a remainder.
Identifying the common factor is an essential first step in factoring polynomials because it helps to simplify the expression and makes further factoring easier.
Consider the expression from the exercise:

• 12\(a^3\)
• -46\(a^2\)
• +40\(a\)

Each of these terms contains the variable \(a\). This means \(a\) is a common factor that can be factored out.
When you factor out \(a\), essentially you are dividing each term in the polynomial by \(a\) and writing down what remains. This gives us the expression: \[a(12a^{2} - 46a + 40)\].
By pulling out the greatest common factor first, you streamline the polynomial and set the stage for further factoring.

A **quadratic expression** is a polynomial of degree 2, typically represented in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
The expression inside the parentheses is a quadratic: \(12a^{2} - 46a + 40\). This type of expression is characterized by its highest exponent being 2.
Understanding the structure of a quadratic expression helps us in using various methods to factor it. These methods include factoring by grouping, using special product formulas, or applying the quadratic formula for solutions.

• Quadratics are everywhere in math, from graphing parabolas to solving equations.
• The simplicity or complexity of factoring a quadratic depends on the coefficients \(a\), \(b\), and \(c\).

In our exercise, the quadratic is already set up nicely for us, falling straightforwardly into the factoring techniques that we’ll discuss next.

**Factoring techniques** are methods used to rewrite a polynomial as a product of its simpler factors, which helps in solving equations more efficiently.
To factor our quadratic \(12a^{2} - 46a + 40\), we can use trial and error or strategic guessing, aided by some rules of thumb.
In this case, the solution has already broken it down to its factored form: \(4a(3a - 5)(a - 2)\). Let's break down how we arrived here:

• Common factoring pulls out anything shared, like the 4 in this case.
• Recognize the factor pairs of the constant that also fit into the quadratic.
• Check by expanding the factors to ensure they combine back into the original polynomial.

These steps are important as they confirm that we've achieved the correct factorization. The ultimate goal of factoring is to simplify polynomials for easier handling, whether solving equations or graphing.