x squared minus 6xy plus 9y squared minus 4z squared

x^2 - 6xy + 9y - 4z^2

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The given expression is:

x^2 - 6xy + 9y^2 - 4z^2

To simplify this expression, you can apply a technique called factoring, which involves breaking down the expression into smaller parts that can be easily dealt with. Let's go through the steps:

Step 1: Recognize the form
The given expression can be recognized as a quadratic expression because it contains variables raised to the power of 2 (x^2, y^2, and z^2).

Step 2: Identify any common factors among the terms
In this case, there are no common factors among the terms.

Step 3: Look for any special factoring forms
The given expression does not appear to be in any special factoring forms, such as perfect square trinomials or difference of squares.

Step 4: Apply the quadratic factoring technique
To factor this quadratic expression, we need to find two binomials whose product equals the original expression.

Let's factor the quadratic expression step by step:

x^2 - 6xy + 9y^2 - 4z^2

Step 4.1: Look for the perfect square trinomial pattern
The first three terms form a perfect square trinomial, which follows the pattern (a - b)^2 = a^2 - 2ab + b^2. In this case, (x - 3y)^2 = x^2 - 6xy + 9y^2.

Step 4.2: Factor out the difference of squares
The last term, -4z^2, can be written as (-2z)^2, which is a perfect square. So, we can represent -4z^2 as (-2z)^2.

At this point, the expression can be factored as:
(x - 3y)^2 - (-2z)^2

Step 4.3: Apply the difference of squares formula
The expression is now in the form a^2 - b^2, which can be factored using the difference of squares formula:
(a - b)(a + b)

Using this formula, we can factor the expression as:
[(x - 3y) - (-2z)][(x - 3y) + (-2z)]

Simplifying further, we have:
(x - 3y + 2z)(x - 3y - 2z)

Therefore, the simplified form of the given expression is (x - 3y + 2z)(x - 3y - 2z).