To simplify the given expression, we need to break it down step-by-step. The expression to simplify is:
[tex]\[
\frac{a^2 - 6ab + 9b^2 - 4c^2}{a - 3b + 2c}
\][/tex]
Let's analyze the numerator first:
[tex]\[
a^2 - 6ab + 9b^2 - 4c^2
\][/tex]
Notice that the first three terms [tex]\(a^2 - 6ab + 9b^2\)[/tex] can be factored as a perfect square:
[tex]\[
a^2 - 6ab + 9b^2 = (a - 3b)^2
\][/tex]
So, the expression now looks like this:
[tex]\[
(a - 3b)^2 - 4c^2
\][/tex]
Next, recognize that the expression [tex]\((a - 3b)^2 - 4c^2\)[/tex] is a difference of squares, which can be factored as:
[tex]\[
(a - 3b)^2 - (2c)^2 = \left((a - 3b) - 2c\right)\left((a - 3b) + 2c\right)
\][/tex]
So, the factored form of the numerator becomes:
[tex]\[
\left(a - 3b - 2c\right)\left(a - 3b + 2c\right)
\][/tex]
Now we can rewrite the entire expression using this factored form:
[tex]\[
\frac{\left(a - 3b - 2c\right)\left(a - 3b + 2c\right)}{a - 3b + 2c}
\][/tex]
We see that the term [tex]\((a - 3b + 2c)\)[/tex] in the numerator and the denominator can be canceled out, provided that [tex]\(a \neq 3b - 2c\)[/tex] to avoid division by zero.
After canceling out the common term, we are left with:
[tex]\[
a - 3b - 2c
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{a - 3b - 2c}
\][/tex]
