What is the difference of squares in the expression 10y² – 4x² – 16y² – x² – 8x – 40 – 25 – 64x² – 48x – 9?

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Understanding the Difference of Squares

The difference of squares is a common algebraic factorization technique that takes the form of a² – b² = (a – b)(a + b). We’ll analyze the expression provided and identify notable patterns that may indicate a difference of squares.

Breaking Down the Expression

The given expression can be stated as:

  • 10y²
  • – 4x²
  • – 16y²
  • – x²
  • – 8x
  • – 40
  • – 25
  • – 64x²
  • – 48x
  • – 9

First, we need to combine like terms and rewrite it:

Expression: 10y² – 16y² – 4x² – x² – 64x² – 48x – 40 – 25 – 9

Combining like terms gives:

  • For y²: 10y² – 16y² = -6y²
  • For x²: -4x² – x² – 64x² = -69x²
  • For x: -48x
  • Constant terms: -40 – 25 – 9 = -74

The expression simplifies to:

-6y² - 69x² - 48x - 74

Identifying the Difference of Squares

We will need to factor out suitable coefficients or expressions. Since none of the terms are in perfect square form, we cannot find an obvious difference of squares.

To apply the difference of squares approach, we have to create terms that fit into the factorization template a² – b², or look for full squares that can be grouped accordingly. Let’s rearrange and regroup the terms:

-69x² - 6y² - 48x - 74

Final Thoughts

The original expression does not naturally lend itself to a difference of squares due to the conflicting signs of the coefficients and lack of perfect squares. However, by rearranging or further grouping terms with complete squares, one could find combinations that fit the difference of squares logic:

To summarize, while we can initially apply the difference of squares, the complexities of the original expression obscure a straightforward application. Diligent algebraic manipulation or transformations may yield different forms or factors.

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