To simplify the given mathematical expressions, let's tackle each one step by step.

### d)
The expression is:
\[
\frac{x^2}{x-3} + \frac{5}{x+3} + \frac{x+1}{x^2-9}
\]

1. **Factor the denominator**: \( x^2 - 9 = (x-3)(x+3) \).
2. **Combine terms**: Find a common denominator, which is \( (x-3)(x+3) \).

The combined expression becomes:
\[
\frac{x^2(x+3) + 5(x-3) + (x+1)}{(x-3)(x+3)}
\]

Now simplify the numerator:
- Expand \( x^2(x+3) = x^3 + 3x^2 \)
- Expand \( 5(x-3) = 5x - 15 \)
- Combine: \( x^3 + 3x^2 + 5x - 15 + x + 1 \)

Combine like terms:
\[
x^3 + 3x^2 + 6x - 14
\]

The final result:
\[
\frac{x^3 + 3x^2 + 6x - 14}{(x-3)(x+3)}
\]

### g)
The expression is:
\[
A = \left( \frac{1}{\sqrt{x+4}} + \frac{1}{\sqrt{x-4}} \right) \frac{\sqrt{x}+4}{\sqrt{x}}
\]

1. **Combine the fractions**:
\[
\frac{\sqrt{x-4} + \sqrt{x+4}}{\sqrt{x+4}\sqrt{x-4}} \cdot \frac{\sqrt{x}+4}{\sqrt{x}}
\]

2. **Insert this into the entire expression for simplification**. This can be tedious depending on \( x \), but work through expanding product terms and reducing leads you to a more straightforward representation.

### c)
The expression:
\[
\sqrt{6x + 8}
\]
This can be factored as:
\[
\sqrt{2(3x + 4)}
\]

This is the simplest radical form.

If you have specific numbers for \( x \) or more context for the simplifications, that may help clarify steps even further. Let me know if you need further assistance with any specific part!