Verified Answer :

  • Remove the parentheses: \(-10h^2 + 8 - 2h - 2\).
  • Group the like terms: \(-10h^2 - 2h + (8 - 2)\).
  • Perform the arithmetic on constant terms: \(8 - 2 = 6\).
  • The final simplified polynomial is \(\boxed{-10h^2 - 2h + 6}\).

Explanation

  1. Analyzing the Problem

The problem asks us to find the sum of two polynomials: \((-10h^2 + 8)\) and \((-2h - 2)\). To simplify this sum, we need to combine like terms—that is, terms that have the same power of the variable \(h\). We start by removing the parentheses from the given expression.

  1. Grouping Like Terms

To add the polynomials, we write the expression without parentheses:
\[(-10h^2 + 8) + (-2h - 2) = -10h^2 + 8 - 2h - 2\]
Now, we group the terms with the same degree together:

  • The quadratic term (\(h^2\)): \(-10h^2\)
  • The linear term (\(h\)): \(-2h\)
  • The constant terms: \(8 - 2\)
  1. Simplifying the Expression

Now, we combine these groups to simplify the expression:

  • The \(h^2\) term remains \(-10h^2\).
  • The \(h\) term remains \(-2h\).
  • The sum of the constant terms is \(8 - 2 = 6\).

Putting it all together, we obtain the expression:
\[-10h^2 - 2h + 6\]
This is the simplified polynomial in standard form.

Examples

Adding and subtracting polynomials is a fundamental skill in algebra used in many fields. For example, if you are calculating the total cost of a project where the material cost is modeled by one polynomial and the labor cost is modeled by another, you would need to add these polynomials to determine the total cost function as a single expression. This helps engineers, economists, and designers simplify complex models into manageable ones.

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