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The distributive property is a fundamental concept in algebra that helps simplify expressions and solve equations. It states that multiplying a sum by a number gives the same result as multiplying each addend individually by the number and then adding the products. In mathematical terms, it can be expressed as: \\[ a(b + c) = ab + ac \]. This property is crucial when dealing with binomials, like in our original exercise.Imagine you were buying 5 bags, each containing 2 apples, and 3 oranges. Rather than counting everything individually, the distributive property lets you multiply 5 by the sum of apples and oranges, streamlining the process.When applying the distributive property to binomials, every term in the first binomial gets multiplied by every term in the second binomial. It lays the groundwork for expanding expressions like \\((2a-b)(-2b+3a)\), ensuring you account for each combination of terms.

Binomial expansion involves multiplying two binomials to express them as a polynomial. The binomial expansion in our exercise follows a common pattern in algebra. By utilizing the distributive property, we multiply each term from the first binomial by each term in the second.In the example given, \\((2a-b)(-2b+3a)\), we expanded it into \\(4\) separate multiplications: \\((2a)(-2b)\), \\((2a)(3a)\), \\((-b)(-2b)\), and \\((-b)(3a)\). This leads us to a fully expanded expression: \\(-4ab + 6a^2 + 2b^2 - 3ab\).By expanding binomials into their polynomial form, you transform a compact multiplication into a clear, step-by-step expression. This method is particularly useful when simplifying complex algebraic expressions before combining like terms.

Like terms are terms in an algebraic expression that have the same variables raised to the same power. Identifying and combining these terms simplifies the expression, making it easier to analyze or solve.In the given example, after expanding the binomials, we had an expression with terms: \\(-4ab\), \\(6a^2\), \\(2b^2\), and \\(-3ab\). Here, the like terms \\(-4ab\) and \\(-3ab\) both contain the product \\(ab\), which can be combined by simply adding their coefficients: \\(-4 - 3 = -7\). This gives the combined term \\(-7ab\).By combining like terms, the polynomial simplifies to the expression \\(6a^2 - 7ab + 2b^2\). This process reduces complexity, allowing you to see the relationship between terms more clearly, and is crucial for solving algebraic equations efficiently.