To simplify the given expression:
[tex]\[
\frac{a^2 - 6ab + 9b^2 - 4c^2}{a - 3b + 2c},
\][/tex]
we start by examining the numerator: [tex]\(a^2 - 6ab + 9b^2 - 4c^2\)[/tex].
First, let's observe if the numerator can be factored.
The terms [tex]\(a^2 - 6ab + 9b^2\)[/tex] suggest a perfect square trinomial:
[tex]\[
a^2 - 6ab + 9b^2 = (a - 3b)^2.
\][/tex]
Next, look at the term [tex]\(-4c^2\)[/tex]. This is a perfect square, which can be written as [tex]\((-2c)^2\)[/tex].
Thus, the numerator can be rewritten as the difference of two squares:
[tex]\[
(a - 3b)^2 - (2c)^2.
\][/tex]
We can now apply the difference of squares formula, [tex]\(x^2 - y^2 = (x - y)(x + y)\)[/tex]:
[tex]\[
(a - 3b)^2 - (2c)^2 = \left( (a - 3b) - 2c \right)\left( (a - 3b) + 2c \right).
\][/tex]
Therefore, our numerator becomes:
[tex]\[
\left( (a - 3b) - 2c \right)\left( (a - 3b) + 2c \right),
\][/tex]
or more clearly:
[tex]\[
(a - 3b - 2c)(a - 3b + 2c).
\][/tex]
The original expression now looks like this:
[tex]\[
\frac{(a - 3b - 2c)(a - 3b + 2c)}{a - 3b + 2c}.
\][/tex]
Since [tex]\(a - 3b + 2c\)[/tex] appears in both the numerator and the denominator, we can cancel it out:
[tex]\[
a - 3b - 2c.
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{a - 3b - 2c}.
\][/tex]
