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The distributive property is one of the foundational principles in algebra, especially when working with expressions that involve multiplication over addition or subtraction. Typically, it is represented by the equation:\[a(b + c) = ab + ac\]
This property essentially involves multiplying a term outside a parenthesis by each term inside the parenthesis and then adding the results. It's like evenly distributing the outer term to each element inside the parentheses.
In the example \((2x + 2)(x - 3)\), the distributive property is applied twice:

• First, multiply \(2x\) by \(x\) and \(-3\).
• Second, multiply \(2\) by \(x\) and \(-3\).

Together, these steps produce the terms \(2x^2 + 2x\) and \(-6x - 6\), which can then be combined by aligning similar elements, leading to the simplified expression.

In algebra, terms that have the same variable and powers are categorized as 'like terms.' Recognizing and combining these terms is crucial for simplifying expressions effectively.For instance, in the expression \(2x^2 + 2x - 6x - 6\), both \(2x\) and \(-6x\) are like terms because they contain the same variable, \(x\), raised to the first power.
This allows us to combine them by adding or subtracting their coefficients:

• Combine \(2x\) and \(-6x\) to get \(-4x\).

Thus, the expression simplifies to \(2x^2 - 4x - 6\).
Understanding like terms helps in streamlining complex algebraic expressions, making calculations simpler and more organized.

Quadratic expressions are polynomial expressions of degree two, represented generally as \(ax^2 + bx + c\). They form the basis for quadratic equations, which graph to create a parabola.In the problem, \(2x^2 - 4x - 6\) presents a quadratic expression, where:

• \(a = 2\)
• \(b = -4\)
• \(c = -6\)

Factors of quadratic expressions often require breaking them into products of linear expressions. This often involves reversing the process of distribution — identifying terms that, when multiplied, return the quadratic expression.
Recognizing these patterns is key to solving or simplifying quadratic expressions, allowing for efficient problem-solving in more complex algebraic tasks.