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The distributive property is a fundamental principle in algebra that allows us to multiply a single term by each term within a parenthesis, thus 'distributing' the multiplication across the terms. It provides a methodical way to simplify and solve more complex equations.

For example, let's consider the expression \( -b(5b^2 + 7b - 35) \). When applying the distributive property, we multiply \( -b \) by each term within the parentheses: \( 5b^2 \) becomes \( -5b^3 \) because you multiply the coefficients (\(-1\) from \( -b \) and \(5\) from \(5b^2\)) and add the exponents (\(1\) from \(b\) and \(2\) from \(b^2\)). Similarly, \( 7b \) becomes \( -7b^2 \) and \( -35 \) becomes \( +35b \)—note here how the negative sign of \( -35 \) turns positive when multiplied by \( -b \) since a negative times a negative results in a positive.

This property is vital because it breaks down complicated expressions into simpler multiplications, making it easier for students to understand and solve polynomial equations.

When dealing with polynomial multiplication, it's essential to understand the structure of monomials and polynomials. A monomial is an algebraic expression with only one term, such as \( -b \) or \( 5b^2 \), and it can include numbers, variables, and exponents. On the other hand, a polynomial is an expression with multiple terms, such as the trinomial \( 5b^2 + 7b - 35 \).

Multiplying a monomial by a polynomial involves using the distributive property to apply the monomial to each term of the polynomial. Take the monomial \( -b \) and the trinomial from the given exercise as an example; each term in the trinomial \( 5b^2 \) (monomial), \( 7b \) (monomial), and \( -35 \) (constant) is multiplied by the monomial \( -b \) to form new monomials, which collectively represent the resulting polynomial after performing the multiplication.

Arithmetic with negative numbers is a cornerstone concept in algebra, especially when multiplying polynomials. Key rules include the fact that multiplying two negative numbers results in a positive number, and multiplying a positive number with a negative one results in a negative number.

For the given exercise, when you multiply \( -b \) with \( -35 \) you get a positive product, resulting in \( +35b \). It’s crucial to remember these rules when performing polynomial multiplication as they affect the sign of each term in the final expression. Incorrectly applying these rules can lead to wrong answers and misunderstandings of algebraic concepts. Therefore, always pay careful attention to the signs of numbers when multiplying terms to ensure accurate results.