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When dealing with algebraic expressions, we often encounter 'like terms.' These are terms that have the same variable raised to the same power. For example, in the expression \(3x^{2} + 4x + 5 + 2x^{2} - x\), the terms \(3x^{2}\) and \(2x^{2}\) are like terms because they both involve \(x^{2}\). To simplify the expression, we combine these like terms by adding or subtracting their coefficients. In this example, combining \(3x^{2}\) and \(2x^{2}\) gives us \(5x^{2}\). Similarly, we combine other like terms such as \(4x - x = 3x\). This step is critical in polynomial subtraction because it helps us to simplify the final result and get a clear, concise expression.

The distributive property is a fundamental concept in algebra that allows us to distribute a single term across terms within parentheses. This is especially useful in polynomial subtraction. When we subtract one polynomial from another, we need to distribute the negative sign across each term of the polynomial being subtracted. For example, consider the operation \(-2x^{2} - 6x + 4 - (9x^{2} - 3x + 7)\). By applying the distributive property, the expression inside the parentheses becomes \(-9x^{2} + 3x - 7\), and thus the entire expression simplifies to \(-2x^{2} - 6x + 4 - 9x^{2} + 3x - 7\). This step ensures that each term of the subtracted polynomial is properly negated, allowing us to accurately combine like terms.

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. In the context of polynomial subtraction, we are dealing specifically with polynomials, which are a type of algebraic expression that consists of terms in the form \(ax^{n}\), where \(a\) is a coefficient and \(n\) is a non-negative integer exponent. Polynomials can have multiple terms, such as \(9x^{2} - 3x + 7\), which has three terms. Understanding algebraic expressions involves recognizing how to manipulate these terms using mathematical operations like addition, subtraction, multiplication, and sometimes division. In our exercise, we subtract two polynomials by writing each term clearly, distributing the negative sign, and then combining like terms to achieve the simplified result.