Verified Answer :

  • Identify the like terms in each expression: the \(c^2\) terms, the \(cd\) terms, and the \(d\) terms.
  • Group the \(c^2\) terms: \(-4c^2 + 8c^2 = 4c^2\).
  • Group the \(cd\) terms: \(7cd + 4cd = 11cd\).
  • Group the \(d\) terms: \(8d - 3d = 5d\).
  • Combine the results to get the sum: \(\boxed{4c^2 + 11cd + 5d}\).

Explanation

  1. Analyzing the Expressions

The task is to find the sum of two algebraic expressions: \((-4c^2 + 7cd + 8d)\) and \((-3d + 8c^2 + 4cd)\). To add these expressions, we identify and group like terms—those that have the same variables and powers. The like terms in these expressions are those containing \(c^2\), \(cd\), and \(d\).

  1. Combining Like Terms

We group the terms as follows:

  • Terms with \(c^2\): \(-4c^2\) and \(8c^2\). Adding these gives: \((-4 + 8)c^2 = 4c^2\).
  • Terms with \(cd\): \(7cd\) and \(4cd\). Adding these gives: \((7 + 4)cd = 11cd\).
  • Terms with \(d\): \(8d\) and \(-3d\). Adding these gives: \((8 - 3)d = 5d\).
  1. Final Simplified Expression

By combining the results from the previous step, the final expression is the sum of the results obtained for each group of like terms:
\[4c^2 + 11cd + 5d\]
Thus, the sum is \(\boxed{4c^2 + 11cd + 5d}\).

Examples

In real-life scenarios, like budgeting or inventory management, you often need to combine different categories of items. For example, if you manage two separate warehouses containing different amounts of items (represented by algebraic variables like \(c\) for crates and \(d\) for delivery boxes), you can add their contents by grouping similar item types together, just like we grouped terms in this algebraic expression to find the total amount.

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