To express $\frac{9x}{(2x+1)(x^2+1)}$ as a sum of partial fractions, we follow these steps:

Step 1: Set up the partial fractions

We assume the partial fraction decomposition takes the form:

$\frac{9x}{(2x+1)(x^2+1)} = \frac{A}{2x+1} + \frac{Bx + C}{x^2+1}$

Step 2: Multiply through by the common denominator

Multiply both sides by $(2x+1)(x^2+1)$ to eliminate the denominators:

$9x = A(x^2 + 1) + (Bx + C)(2x + 1)$

Step 3: Expand and collect like terms

Expand the right-hand side:

$9x = A(x^2 + 1) + (Bx + C)(2x + 1)$

Expand each term:

$9x = A(x^2) + A(1) + Bx(2x) + Bx(1) + C(2x) + C(1)$

$9x = Ax^2 + A + 2Bx^2 + Bx + 2Cx + C$

Now, collect like terms:

$9x = (A + 2B)x^2 + (B + 2C)x + (A + C)$

Step 4: Match coefficients

Now, match the coefficients of corresponding powers of $x$ on both sides:

For $x^2$: $A + 2B = 0$

For $x$: $B + 2C = 9$

For the constant term: $A + C = 0$

Step 5: Solve the system of equations

We now solve this system of equations:

1. From $A + 2B = 0$, we have $A = -2B$.
2. Substitute $A = -2B$ into $A + C = 0$ to get $-2B + C = 0$, so $C = 2B$.
3. Substitute $C = 2B$ into $B + 2C = 9$ to get $B + 4B = 9$, so $5B = 9$, which gives $B = \frac{9}{5}$.
4. Since $C = 2B$, we get $C = 2 \times \frac{9}{5} = \frac{18}{5}$.
5. Since $A = -2B$, we get $A = -2 \times \frac{9}{5} = -\frac{18}{5}$.

Step 6: Write the final partial fraction decomposition

Now that we have $A$, $B$, and $C$, substitute them back into the partial fraction form:

$\frac{9x}{(2x+1)(x^2+1)} = \frac{-\frac{18}{5}}{2x+1} + \frac{\frac{9}{5}x + \frac{18}{5}}{x^2+1}$

This can be simplified to:

$\frac{9x}{(2x+1)(x^2+1)} = -\frac{\frac{18}{5}}{2x+1} + \frac{\frac{9}{5}(x+2)}{x^2+1}$

This is the partial fraction decomposition of the given expression.

Let me know if you need any further details or have any questions.

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Here are five related questions:

1. How do you perform partial fraction decomposition when dealing with repeated factors?
2. Can partial fractions be applied to improper fractions? If so, how?
3. What are the applications of partial fractions in solving integrals?
4. How do you solve a system of equations with three or more variables?
5. How do partial fractions help in simplifying complex rational functions?

Tip: When solving partial fraction problems, always double-check your coefficients and expand the terms to ensure the decomposition is correct.