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Binomial expansion involves multiplying a binomial by itself or another polynomial and expressing the result as a polynomial. In the provided exercise, the binomial \(x+5\) is expanded against a trinomial \(2x^2-3x+1\). This process involves systematically multiplying each term in the first polynomial with each term in the second polynomial. The key to successful binomial expansion is understanding that every term in the binomial must be multiplied by every term in the trinomial.

For example, when \(x\) from the binomial is multiplied by each of the trinomial’s three terms, the outcome is \(2x^3 - 3x^2 + x\), representing the expansion of \(x\) across the trinomial. The same process is followed by multiplying the 5 in \(x+5\) with each part of the trinomial. The concept of binomial expansion essentially applies the distributive property, where each element in the binomial is distributed across the terms of the trinomial, then simplified by combining like terms to yield a final polynomial expression.

Understanding the distributive property is crucial in polynomial multiplication. This property states that a term multiplied with a sum or difference of terms can be broken down into individual multiplications, i.e., \(a(b + c) = ab + ac\). The distributive property makes it easier to expand polynomials, especially when dealing with expressions like the binomial and trinomial in our exercise.

In Step 1 of the solution, \(x\) is distributed across the trinomial terms by multiplying it with each term, resulting in \(2x^3 - 3x^2 + x\). Similarly, Step 2 involves distributing the constant 5 in the same manner to get \(10x^2 - 15x + 5\). By using the distributive property, complex multiplication can be broken down into simpler, manageable parts, which are later combined to find the complete product.

After the distributive property has been used to expand the polynomial, the next step is to simplify the expression by combining like terms. Like terms are terms within an algebraic expression that have the same variable raised to the same power. In our example, once we distribute and expand our terms, we end up with several different terms. The goal is to merge terms that have the same variables and exponents.

In Step 4, the expression \(2x^3 - 3x^2 + x\) and \(10x^2 - 15x + 5\) derived from the previous steps contain like terms that need to be combined. For instance, \( -3x^2\) and \(10x^2\) are like terms, as are \(x\) and \( -15x\). Thus, they are added together to give \(2x^3 + 7x^2 - 14x + 5\), which is the simplified form of the expanded polynomial. This process of combining like terms simplifies the expression and is a crucial step in solving polynomial multiplication.