Simplifying Polynomial Expressions
In mathematics, simplifying expressions is a fundamental skill. This involves combining like terms and performing operations to make the expression more concise. Let's explore how to simplify the expression:
(5b  6b^3 + 2b^4)  (9b^3 + 4b^4  7)
Understanding the Expression
Before we start simplifying, let's break down the expression:
 Parentheses: The expression has two sets of parentheses, indicating that we need to consider the order of operations.
 Terms: The expression is made up of several terms, each consisting of a coefficient and a variable raised to a power. For example, 5b is a term with a coefficient of 5 and a variable 'b' raised to the power of 1.
 Like Terms: Terms with the same variable raised to the same power are considered like terms. For example, 6b^3 and 9b^3 are like terms.
Simplifying the Expression

Distribute the negative sign: Since we have a minus sign in front of the second set of parentheses, we need to distribute it to each term inside the parentheses. This changes the signs of all the terms within the parentheses:
(5b  6b^3 + 2b^4)  (9b^3 + 4b^4  7) = 5b  6b^3 + 2b^4  9b^3  4b^4 + 7

Combine like terms: Now we can combine the terms with the same variable and exponent.
 b^4 terms: 2b^4  4b^4 = 2b^4
 b^3 terms: 6b^3  9b^3 = 15b^3
 b terms: 5b
 Constant terms: + 7

Rewrite the simplified expression:
2b^4  15b^3 + 5b + 7
Final Result
Therefore, the simplified form of the given expression (5b  6b^3 + 2b^4)  (9b^3 + 4b^4  7) is 2b^4  15b^3 + 5b + 7.