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The image shows two rational expressions with operations that require simplification. Let's break them down and solve them step by step.

### Part (a):
\[
\frac{2x - 1}{x + 3} + \frac{x}{2x + 1}
\]

#### Step 1: Find the Least Common Denominator (LCD)
The denominators are \(x + 3\) and \(2x + 1\), so the least common denominator (LCD) is \((x + 3)(2x + 1)\).

#### Step 2: Rewrite each fraction with the LCD
We rewrite each fraction with the common denominator:

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

#### Step 3: Add the numerators
Now that the denominators are the same, we add the numerators:

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

Simplify the numerator:

\[
(2x - 1)(2x + 1) = 4x^2 - 1
\]
\[
x(x + 3) = x^2 + 3x
\]

So the numerator becomes:

\[
4x^2 - 1 + x^2 + 3x = 5x^2 + 3x - 1
\]

#### Final expression for part (a):
\[
\frac{5x^2 + 3x - 1}{(x + 3)(2x + 1)}
\]

### Part (b):
\[
\frac{2}{x^4 + 1} - \frac{1}{x}
\]

#### Step 1: Find the Least Common Denominator (LCD)
The denominators are \(x^4 + 1\) and \(x\). The least common denominator is \(x(x^4 + 1)\).

#### Step 2: Rewrite each fraction with the LCD
We rewrite each fraction with the common denominator:

\[
\frac{2}{x^4 + 1} = \frac{2x}{x(x^4 + 1)}
\]
\[
\frac{1}{x} = \frac{x^4 + 1}{x(x^4 + 1)}
\]

#### Step 3: Subtract the numerators
Now that the denominators are the same, we subtract the numerators:

\[
\frac{2x - (x^4 + 1)}{x(x^4 + 1)}
\]

Simplify the numerator:

\[
2x - x^4 - 1
\]

#### Final expression for part (b):
\[
\frac{-x^4 + 2x - 1}{x(x^4 + 1)}
\]

---

### Summary of Final Answers:
- Part (a): \(\frac{5x^2 + 3x - 1}{(x + 3)(2x + 1)}\)
- Part (b): \(\frac{-x^4 + 2x - 1}{x(x^4 + 1)}\)

Would you like further details or have any questions?

### Related Questions:
1. How do you factor complex expressions with quadratic terms?
2. How do you solve rational equations involving polynomials?
3. What are the steps to simplify complex rational expressions?
4. Can we apply partial fraction decomposition in these cases?
5. How can you check your final solution for rational expressions?

### Tip:
Always find the least common denominator (LCD) before adding or subtracting rational expressions to ensure you simplify the expressions correctly.

Perform the indicated operation and simplify: (2x - 1) / (x + 3) + x / (2x + 1) and 2 / (x^4 + 1) - 1 / x.

Learn how to perform and simplify rational expressions like (2x - 1) / (x + 3) + x / (2x + 1) and 2 / (x^4 + 1) - 1 / x, using the least common denominator and algebraic simplifications. This problem covers core algebraic techniques.

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