Let's simplify the expression:
[tex]\[
\frac{x+3}{x^2+3x+9} + \frac{x-3}{x^2-3x+9} - \frac{54}{x^2 + 9x^2 + 82}
\][/tex]
First, notice that we have three separate fractions. Our goal is to simplify each part and then combine them.
1. Simplify the first term:
[tex]\[
\frac{x+3}{x^2 + 3x + 9}
\][/tex]
The denominator [tex]\(x^2 + 3x + 9\)[/tex] can't be factored further in a simple way, so we'll leave it as is for now.
2. Simplify the second term:
[tex]\[
\frac{x-3}{x^2 - 3x + 9}
\][/tex]
Similarly, the denominator [tex]\(x^2 - 3x + 9\)[/tex] can't be factored further in a simple way.
3. Simplify the third term:
[tex]\[
\frac{54}{x^2 + 9x^2 + 82}
\][/tex]
First, simplify the denominator:
[tex]\[
x^2 + 9x^2 = 10x^2, \quad \therefore \quad x^2 + 9x^2 + 82 = 10x^2 + 82
\][/tex]
So the third term becomes:
[tex]\[
\frac{54}{10x^2 + 82}
\][/tex]
Now, to combine these fractions, let's first attempt to add [tex]\(\frac{x+3}{x^2+3x+9} + \frac{x-3}{x^2-3x+9}\)[/tex].
Since the denominators are not the same, finding a common denominator might not simplify the expression easily.
After examining simplification steps and common factors (detailed algebraic manipulations), the overall expression simplifies to the following fraction by combining all terms and simplifying them:
[tex]\[
\frac{10x^5 - 27x^4 + 82x^3 - 243x^2 - 2187}{5x^6 + 86x^4 + 774x^2 + 3321}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{10x^5 - 27x^4 + 82x^3 - 243x^2 - 2187}{5x^6 + 86x^4 + 774x^2 + 3321}
\][/tex]
This is the detailed and simplified form of the given mathematical question.
