Factorizing the Expression (x² - 6xy + 9y²) - 25

We are asked to factorize the given expression: $(x^2 - 6xy + 9y^2) - 25$.

Let's look at the first part of the expression, $(x^2 - 6xy + 9y^2)$. This looks like a perfect square trinomial. Recall the algebraic identity for a perfect square trinomial:

• $(a - b)^2 = a^2 - 2ab + b^2$

Comparing $(x^2 - 6xy + 9y^2)$ with $a^2 - 2ab + b^2$, we can identify $a$ and $b$:

• $a^2 = x^2 \implies a = x$
• $b^2 = 9y^2 \implies b = 3y$
• Check the middle term: $-2ab = -2(x)(3y) = -6xy$. This matches the middle term in the given expression.

Therefore, the first part can be written as a perfect square:

$(x^2 - 6xy + 9y^2) = (x - 3y)^2$

Now, substitute this back into the original expression:

$(x^2 - 6xy + 9y^2) - 25 = (x - 3y)^2 - 25$

The expression is now in the form of a difference of squares. Recall the algebraic identity for the difference of squares:

• $A^2 - B^2 = (A - B)(A + B)$

In our expression, $(x - 3y)^2 - 25$, we can identify $A$ and $B$:

• $A^2 = (x - 3y)^2 \implies A = (x - 3y)$
• $B^2 = 25 \implies B = \sqrt{25} = 5$

Now, apply the difference of squares formula with $A = (x - 3y)$ and $B = 5$:

$A^2 - B^2 = (A - B)(A + B)$

$(x - 3y)^2 - 5^2 = ((x - 3y) - 5)((x - 3y) + 5)$

Simplifying the terms inside the parentheses:

$(x - 3y - 5)(x - 3y + 5)$

This is the factored form of the expression $(x^2 - 6xy + 9y^2) - 25$.

Comparing this factored form with the given options, we find that it matches option 4:

• $(x - 3y - 5)(x - 3y + 5)$

Thus, the factorization of $(x^2 - 6xy + 9y^2) - 25$ is $(x - 3y - 5)(x - 3y + 5)$.