Verified Answer :

  • Remove the parentheses by distributing the negative sign: \(-2x^2 + 9x + 10 - 5x - 8\).
  • Group like terms: \(-2x^2 + (9x - 5x) + (10 - 8)\).
  • Simplify the expression: \(\boxed{-2x^2 + 4x + 2}\).

Explanation

  1. Understanding the Problem

To solve the expression \((-2x^2 + 9x + 10) - (5x + 8)\), we first identify that this is a subtraction of two polynomials. We have a quadratic polynomial \(-2x^2 + 9x + 10\) and a linear polynomial \(5x + 8\). Our task is to perform the subtraction by distributing the negative sign to all terms within the parentheses of the second polynomial and then grouping and combining like terms.

  1. Distributing the Negative Sign

To subtract the second polynomial from the first, we remove the parentheses. When we subtract \((5x + 8)\), we distribute the negative sign:
\[-(5x + 8) = -5x - 8\]
Now, the entire expression becomes:
\[-2x^2 + 9x + 10 - 5x - 8\]

  1. Combining Like Terms

Next, we group the terms with the same variable powers (like terms) together:
\[-2x^2 + (9x - 5x) + (10 - 8)\]
Now, we perform the arithmetic for each group:

  • For the \(x^2\) term, we have \(-2x^2\).
  • For the \(x\) terms, we calculate \(9x - 5x = 4x\).
  • For the constant terms, we calculate \(10 - 8 = 2\).

Combining these results, we get the simplified expression:
\[-2x^2 + 4x + 2\]

Examples

Algebraic expressions like this are used in physics and engineering to model motion and optimization. For instance, if you have an initial profit model represented by \(-2x^2 + 9x + 10\) and a cost function of \(5x + 8\), subtracting the costs from the profit tells you the net profit model: \(-2x^2 + 4x + 2\). Understanding how to combine like terms allows you to simplify complex physical or financial models into manageable functions.

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