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The distributive property is a fundamental algebraic principle used to multiply expressions. In simple terms, it means spreading out the multiplication over addition or subtraction. When you multiply a polynomial by another polynomial, you apply the distributive property multiple times. Each term in the first polynomial needs to be multiplied by each term in the second polynomial. For example, in the given exercise \(\left(5 x^{2}+2 x+1\right)\left(x^{2}-3 x+5\right)\), you multiply each term from \(\left(5 x^{2}+2 x+1\right)\) by each term in \(\left(x^{2}-3 x+5\right)\). So, \(5 x^{2} \cdot x^{2}\), \(5 x^{2} \cdot (-3 x)\), and \(5 x^{2} \cdot 5\) are examples of applications of the distributive property. This creates a broader expression which we need to simplify in the next steps.

Once you have applied the distributive property, you have a large number of terms. The next step is to simplify by combining like terms. Like terms are terms that have the same variable raised to the same power. In the given problem, after distributing, we get terms like \(5 x^{4} + (-15 x^{3}) + 25 x^{2} + 2 x^{3} + (-6 x^{2}) + 10 x + x^{2} + (-3 x) + 5\). To combine like terms, you add or subtract the coefficients in front of the variables with the same exponent. For example, terms \(5 x^{4}\) stand alone because there are no other \(x^{4}\) terms. For the \(x^{3}\) terms: \(-15 x^{3} + 2 x^{3} = -13 x^{3}\). This combining of like terms simplifies our expression.

Polynomial simplification is the process of reducing a polynomial to its simplest form. It involves using the distributive property to expand the expression and then combining like terms as described. In the example \(\left(5 x^{2}+2 x+1\right)\left(x^{2}-3 x+5\right)\), after distributing and combining like terms, the expanded polynomial \(5 x^{4} - 13 x^{3} + 20 x^{2} + 7 x + 5\) is simplified as much as possible. Simplification makes it easier to understand the behavior of the polynomial and to perform further algebraic operations. Always ensure that all like terms are combined and the expression is written in standard form, which means ordering terms from the highest to the lowest degree.