) Simplify: (3x+1)/(9x^2+3x+1)+(3x-1)/(3x^2-3x+1)+2/(81x^4+9x^2+1) 		

We need to simplify the expression:

\[
\frac{3x + 1}{9x^2 + 3x + 1} + \frac{3x - 1}{3x^2 - 3x + 1} + \frac{2}{81x^4 + 9x^2 + 1}
\]

### Step 1: Factor the Denominators (if possible)

#### First Denominator: \( 9x^2 + 3x + 1 \)
We can check if this quadratic is factorable using the quadratic formula or by inspection. The discriminant is \( \Delta = (3)^2 - 4(9)(1) = 9 - 36 = -27 \), which means this quadratic is not factorable with real numbers.

#### Second Denominator: \( 3x^2 - 3x + 1 \)
Similarly, check the discriminant: \( \Delta = (-3)^2 - 4(3)(1) = 9 - 12 = -3 \), which also cannot be factored using real numbers.

#### Third Denominator: \( 81x^4 + 9x^2 + 1 \)
This expression is a perfect square trinomial:
\[
81x^4 + 9x^2 + 1 = (9x^2 + 1)^2
\]
It factors as \( (9x^2 + 1)^2 \).

### Step 2: Combine the Fractions

To add these fractions, we need a common denominator. The common denominator will be \( (9x^2 + 1)^2 \), because the first two denominators already have terms that can fit into this expression.

- The first fraction \( \frac{3x + 1}{9x^2 + 3x + 1} \) has a denominator that can be expressed as \( 9x^2 + 3x + 1 \), which is part of \( (9x^2 + 1)^2 \), so we multiply the numerator and denominator by \( 9x^2 + 1 \).
- The second fraction \( \frac{3x - 1}{3x^2 - 3x + 1} \) can similarly be modified to have the same denominator.
- The third fraction \( \frac{2}{(9x^2 + 1)^2} \) already has the required denominator.

Thus, the expression becomes:

\[
\frac{(3x + 1)(9x^2 + 1)}{(9x^2 + 1)^2} + \frac{(3x - 1)(9x^2 + 1)}{(9x^2 + 1)^2} + \frac{2}{(9x^2 + 1)^2}
\]

### Step 3: Simplify the Numerators

Now we simplify the numerators:

1. \( (3x + 1)(9x^2 + 1) = 27x^3 + 3x + 9x^2 + 1 \)
2. \( (3x - 1)(9x^2 + 1) = 27x^3 - 3x + 9x^2 - 1 \)

Now add the numerators:

\[
(27x^3 + 3x + 9x^2 + 1) + (27x^3 - 3x + 9x^2 - 1) + 2
\]

Combine like terms:

\[
54x^3 + 18x^2 + 2
\]

### Step 4: Final Expression

The simplified expression is:

\[
\frac{54x^3 + 18x^2 + 2}{(9x^2 + 1)^2}
\]

### Final Answer:
\[
\boxed{\frac{54x^3 + 18x^2 + 2}{(9x^2 + 1)^2}}
\]

Would you like more details on any of these steps or have any questions?

Here are 5 related questions for further practice:
1. Simplify \( \frac{2x+3}{x^2+3x+2} + \frac{3x-1}{x^2+5x+6} \).
2. Factor \( 81x^4 + 9x^2 + 1 \) step-by-step.
3. Simplify \( \frac{4x+5}{x^2+5x+6} - \frac{3x-2}{x^2+3x+2} \).
4. Simplify \( \frac{x}{x^2+2x+1} + \frac{1}{x+1} \).
5. Simplify \( \frac{2}{x^2 - 4} + \frac{3}{x^2 - 9} \).

**Tip**: Always look for common factors and perfect squares when simplifying expressions involving polynomials.

(3x+1)/(9x^2+3x+1) + (3x-1)/(3x^2-3x+1) + 2/(81x^4+9x^2+1)

Simplify the rational expression (3x+1)/(9x^2+3x+1) + (3x-1)/(3x^2-3x+1) + 2/(81x^4+9x^2+1) step-by-step. This problem involves factoring quadratics, finding a common denominator, and simplifying the numerator. Suitable for students in Grades 9-11, with an emphasis on algebra and rational expressions.

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