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Factoring quadratics is an important skill in simplifying algebraic expressions. Quadratic expressions are polynomials of degree two, typically in the form of \(ax^2 + bx + c\). Factoring involves breaking down these expressions into a product of simpler terms known as factors. For example, in the expression \(3x^2 - 4x\), we notice that the term \(x\) is common in both parts of the expression.
To factor out \(x\), divide each term by \(x\) and rewrite the expression:

• The first term \(3x^2\) becomes \(3x\).
• The second term \(-4x\) simplifies to \(-4\).

Putting it together, the factored form is \(x(3x - 4)\). Factoring quadratics is often the first key step in simplifying more complex expressions involving rational expressions.

Rational expressions are fractions that have polynomials in both the numerator and the denominator. Simplifying these expressions involves reducing them to their simplest form, much like simplifying numerical fractions. A reduction is accomplished when you cancel out common terms both in the numerator and the denominator.
In our exercise, we start with the expression \(\frac{2x^2}{3x^2 - 4x}\), which is a rational expression. One of our first tasks was to factor the denominator to reveal any common factors it shares with the numerator. This helps us in the next step: canceling common factors to shorten and simplify the fraction. Observing the structure of rational expressions closely allows us to identify these shared factors and simplify efficiently.

Once we factor the necessary parts of an expression, we can cancel out common factors, simplifying the expression to its most reduced form. Canceling common factors works in algebra just as it does in arithmetic fractions. It's important to note that cancellation only applies to multiplication; hence, a common term must be a factor, not just an addend, to be canceled.
In the given example, individual terms \(x\) in both the numerator \(2x^2\) and the denominator \(x(3x - 4)\) can be canceled. This cancellation arises from dividing both the numerator and the denominator by their shared factor \(x\).

• The numerator \(2x^2\) becomes \(2x\) after cancellation.
• The denominator transforms from \(x(3x - 4)\) to \(3x - 4\).

Taking the time to factor a problem first ensures that we are active in simplifying the expression by canceling only appropriate terms, leading to a reduced form \(\frac{2x}{3x - 4}\).