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Understanding the distributive property is essential when dealing with polynomial multiplication. This property allows us to multiply a single term outside of a parenthesis by each term inside the parenthesis. For our exercise involving \( (5x^2-2)(x-1) \), the distributive property is applied by multiplying each term in the first polynomial by each term in the second polynomial.

Let's see it in action. Initially, the term \( 5x^2 \) is distributed over \( x \) and \( -1 \), resulting in \( 5x^2 \cdot x \) and \( 5x^2 \cdot -1 \). This is how we get \( 5x^3 \) and \( -5x^2 \). Similarly, the term \( -2 \) is then distributed over the same terms producing \( -2x \) and \( +2 \). This methodical approach ensures that all terms are accounted for in the final expression.

Once we have distributed the terms as outlined in the distributive property, we often find that there are similar terms in the resulting expression—these are our 'like terms.' Combining like terms involves adding or subtracting coefficients of these terms while keeping their variable parts unchanged.

In our example, after distribution, we get \( 5x^3 - 5x^2 -2x + 2 \). Here there are no like terms to combine, which can sometimes be the case. Like terms have the same variable to the same power. For example, \( 2x^2 \) and \( 3x^2 \) are like terms, while \( 2x^2 \) and \( 3x \) are not. It is important to understand that like terms can only be combined with each other, which simplifies the expression and makes it easier to understand.

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. In the exercise we're examining, \( (5x^2-2)(x-1) \) is an example of a polynomial expression, specifically a binomial times a binomial.

These expressions can get quite complex, and each part has a specific name. The number in front of the variable (like the 5 in \( 5x^2 \) ) is called the coefficient. Variables (x) can be raised to a power, like \( x^2 \) which denotes that x is squared. When working with algebraic expressions, recognizing the structure and types of polynomials can provide insight into the most efficient methods for simplifying them, as we've applied with the distributive property and combining like terms.