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When faced with an expression that contains parentheses, one of the first steps you'll want to take is distributing any coefficients. Distributing a coefficient means you multiply it by each term inside the parentheses. This helps to simplify or eliminate the parentheses altogether.
For example, consider the expression \(2(b+c)\). Here, you distribute the 2 by multiplying it with both \(b\) and \(c\), resulting in \(2b + 2c\).
This method follows the Distributive Property of multiplication over addition, which states: \[a(b + c) = ab + ac\] It’s a reliable way to ensure that every term inside the bracket is accounted for when simplifying an expression. In the original exercise, the coefficients 2 and -4 were distributed across their respective terms, transforming the expression significantly.

Combining like terms is the next critical step after distributing coefficients. Like terms are terms in an algebraic expression that have the exact same variables raised to the same power.
To simplify expressions, you sum their coefficients. For instance, terms like \(b + 2b\) are like terms. Since both contain the variable \(b\), you simply add 1 (the implicit coefficient of \(b\)) and 2 to get \(3b\). A useful tip is to circle or highlight like terms, which can help prevent errors and ensure nothing is overlooked. In the given problem, identifying and combining terms like \(2c\) and \(-4c\) results in \(-2c\), further streamlining the expression.

The final goal when working with algebraic expressions is simplification. Simplification means transforming the expression into its simplest form without changing its value.
After distributing coefficients and combining like terms, what's left should be a succinct expression that’s easier to understand and use.
For the example given, the original expression \( (3a+b)+2(b+c)-4(c+d) \) was simplified with distributed coefficients and combined like terms into \(3a + 3b - 2c - 4d\). This simplified form is much more manageable, setting the stage for further algebraic operations or for drawing quick insights, making it essential for solving or evaluating equations in homework and exams.