3/(x^2-9)-x/(x^2-6x+9)+1/(x+3)

3/((x+3)(x-3))- x/(x-3)(x-3) + 1/(x+3)

The LCD is (x-3)(x-3)(x+3)

= (3(x-3) - x(x+3) + 1/((x-3)(x-3))/((x-3)(x-3)(x+3))
= ....
take over

Hmm. I get

(3(x-3) - x(x+3) + 1(x-3)(x-3)) / ((x-3)(x-3)(x+3))

To simplify the expression 3/(x^2-9) - x/(x^2-6x+9) + 1/(x+3), we need to find a common denominator for all three fractions.

The denominators in the three fractions are (x^2-9), (x^2-6x+9), and (x+3). To find the common denominator, we need to factorize these expressions.

(x^2-9) = (x+3)(x-3)
(x^2-6x+9) = (x-3)(x-3) = (x-3)^2

Now, we can rewrite the expression using the common denominator:

[3/(x+3)(x-3)] - [x/(x-3)^2] + 1/(x+3)

Next, simplify each fraction:

3/(x+3)(x-3) - x/(x-3)^2 + 1/(x+3)

To subtract fractions, we need to have the same denominator. So, we need to rewrite the first fraction with the denominator of (x-3)^2.

To do this, we multiply the numerator and denominator of the first fraction by (x-3):

[3(x-3)/(x+3)(x-3)(x-3)] - x/(x-3)^2 + 1/(x+3)

Simplifying the first fraction:

[3x - 9/(x+3)(x-3)^2] - x/(x-3)^2 + 1/(x+3)

Now, we have a common denominator for all three fractions: (x-3)^2 * (x+3).

Combine the numerators:

[3x - 9 - x(x+3)] / [(x+3)(x-3)^2] + 1/(x+3)

Simplify and distribute:

(3x - 9 - x^2 - 3x) / [(x+3)(x-3)^2] + 1/(x+3)

Combine like terms:

(-x^2) / [(x+3)(x-3)^2] + 1/(x+3)

Now, let's simplify further:

-x^2 / [(x+3)(x-3)^2] + 1/(x+3)

To add or subtract fractions, we need to have the same denominator:

(-x^2 + (x-3)^2) / [(x+3)(x-3)^2]

And we can simplify the numerator:

(-x^2 + x^2 - 6x + 9) / [(x+3)(x-3)^2]

Simplify:

(-6x + 9) / [(x+3)(x-3)^2]

Therefore, the simplified expression is (-6x + 9) / [(x+3)(x-3)^2].