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The distributive property is a fundamental rule in algebra used for multiplying expressions. It allows you to distribute each term in one expression to every term in another. When you have a polynomial expression such as \((a+b)(c+d)\), you multiply every term in the first parenthesis by every term in the second parenthesis. Here's a quick breakdown of how this works:

• Multiply the first term in the first parenthesis by each term in the second parenthesis. For example, \(a \times c\) and \(a \times d\).
• Next, do the same with the second term in the first parenthesis, so \(b \times c\) and \(b \times d\).

This way, you ensure that every combination of terms is included in the final expanded expression. In the case of \((-2b^2 - 3b + 4)(b - 5)\), each term from the trinomial \((-2b^2, -3b, 4)\) is distributed across the binomial \(b - 5\), resulting in six terms that need to be simplified and combined to find the overall product of the original expression.

Combining like terms is a method used to simplify polynomials by merging terms with the same variable and exponent. After distributing the terms, you're often left with a long list of terms that need simplification. Here's how you can master this step:

• Identify like terms: Look for terms that involve the same power of the same variable. In the example \(-2b^3 + 10b^2 - 3b^2 + 15b + 4b - 20\), identify \(10b^2\) and \(-3b^2\) as like terms because they both have \("b"\) squared.
• Add or subtract the coefficients: Simply add or subtract the numerical coefficients of these terms. For example, for \((10b^2 - 3b^2)\), just calculate \(10 - 3\) to get \(7b^2\).

By combining like terms, you reduce the expression to its simplest form. This makes it easier to interpret and solve any further operations.

Simplifying expressions is the process of reducing an algebraic expression to its simplest form. This often involves applying both the distributive property and combining like terms as discussed:

• Start by distributing the terms, as explained in the distributive property section. You'll get a mix of terms you've expanded from the given polynomial expressions.
• Then, combine like terms as described. This step reduces the number of terms you have to deal with.

Once these steps are completed, you will have a simplified expression that retains the same value as the original but is much easier to work with. In the example \(-2b^3 + 10b^2 - 3b^2 + 15b + 4b - 20\), combining and simplifying within each grouped set of like terms results in the final expression \-2b^3 + 7b^2 + 19b - 20\. Simplifying expressions is crucial for solving equations efficiently and accurately in mathematics.