In Exercises 65–92, factor completely, or state that the polynomial is prime. x²−10x+25−36y²

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Identify the structure of the polynomial: The given expression is \( x^2 - 10x + 25 - 36y^2 \). Notice that \( x^2 - 10x + 25 \) is a quadratic trinomial, and \( 36y^2 \) is a perfect square.

Factor \( x^2 - 10x + 25 \): Recognize that \( x^2 - 10x + 25 \) is a perfect square trinomial. It can be factored as \( (x - 5)^2 \), since \( (x - 5)(x - 5) = x^2 - 10x + 25 \).

Rewrite the expression: Substitute \( (x - 5)^2 \) for \( x^2 - 10x + 25 \). The expression becomes \( (x - 5)^2 - 36y^2 \).

Recognize the difference of squares: The expression \( (x - 5)^2 - 36y^2 \) is a difference of squares, which can be factored using the formula \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a = (x - 5) \) and \( b = 6y \).

Factor the difference of squares: Apply the formula to factor \( (x - 5)^2 - 36y^2 \) as \( ((x - 5) - 6y)((x - 5) + 6y) \).

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