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When expanding expressions like \((x-11)(3x+2)\), we use a fundamental mathematical principle known as the Distributive Property. This property allows us to break down complex multiplication problems into simpler parts. Specifically, the Distributive Property states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. Mathematically, this is expressed as:

• \(a(b+c) = ab + ac\)

In our exercise, \((x-11)\) and \((3x+2)\) are two binomials. To expand them, we multiply each term from \((x-11)\) by each term from \((3x+2)\). This step-by-step multiplication, also called distributing, results in four products:

• \(x \cdot 3x\)
• \(x \cdot 2\)
• \(-11 \cdot 3x\)
• \(-11 \cdot 2\)

Using the Distributive Property correctly ensures that no term is left out, laying the groundwork for simplifying the expression further.

Once we have distributed and expanded an expression, like in the exercise where we derived \(3x^2 + 2x - 33x - 22\), the next important step is combining like terms. Like terms are terms that contain the same variables raised to the same power. Only the numerical coefficients differ.

In our example, the terms \(2x\) and \(-33x\) are like terms because they both contain the variable \(x\). To combine them, simply add their coefficients:

• \(2x + (-33x) = -31x\)

This process of combining simplifies the expanded polynomial, resulting in fewer terms and makes the expression easier to manage. After combining, the expression becomes \(3x^2 - 31x - 22\), where all like terms have been merged into single terms.

A quadratic expression is a polynomial where the highest degree of the variable is squared, or raised to the power of two. In algebra, these expressions often appear in the standard form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.

The exercise we solved resulted in \(3x^2 - 31x - 22\), which is a classic example of a quadratic expression. The term \(3x^2\) denotes it as quadratic, and it's crucial because it influences the shape and direction of the quadratic graph if plotted. The coefficient \(3\) indicates how wide or narrow the parabola is.

• The \(x^2\) term gives the polynomial its name: quadratic.
• The linear term \(-31x\) impacts the slope or tilt of the graph.
• The constant term \(-22\) affects the position of the graph's y-intercept.

Understanding quadratic expressions is essential in solving many mathematical problems, from graphing parabolas to finding roots through factoring or using the quadratic formula.