To solve the given expression \( A \) and its parts, we can go through the following steps:

### a) Rút gọn biểu thức A.

Given:

\[
A = \left( \frac{x}{x+3} + \frac{3-x}{x+3} \cdot \frac{x^2+3x+9}{x^2-9} \right) : \frac{3}{x+3}
\]

#### Step 1: Simplify the expression inside the parenthesis:

Combine the terms within the parenthesis (use a common denominator):

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

So the expression becomes:

\[
A = \left( \frac{3}{x+3} \cdot \frac{x^2+3x+9}{x^2-9} \right) : \frac{3}{x+3}
\]

#### Step 2: Simplify the division:

Dividing by a fraction is the same as multiplying by its reciprocal:

\[
A = \frac{3}{x+3} \cdot \frac{x^2+3x+9}{x^2-9} \cdot \frac{x+3}{3}
\]

Cancel out the common terms:

\[
A = \frac{x^2+3x+9}{x^2-9}
\]

#### Step 3: Simplify the fraction:

Notice \(x^2 - 9 = (x - 3)(x + 3)\):

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

### b) Tìm giá trị của A tại \( x = -\frac{1}{2} \).

Substitute \( x = -\frac{1}{2} \) into the simplified expression of \( A \):

\[
A = \frac{\left(-\frac{1}{2}\right)^2 + 3\left(-\frac{1}{2}\right) + 9}{\left(-\frac{1}{2} - 3\right)\left(-\frac{1}{2} + 3\right)}
\]

Calculate each part:

\[
= \frac{\frac{1}{4} - \frac{3}{2} + 9}{\left(-\frac{7}{2}\right)\left(\frac{5}{2}\right)}
\]

Simplify:

\[
= \frac{\frac{37}{4}}{\left(-\frac{35}{4}\right)} = -\frac{37}{35}
\]

### c) Tìm giá trị của \( x \) để \( A < 0 \).

We know:

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

For \( A \) to be negative:

1. The numerator and denominator have opposite signs.

Determine the sign of the denominator:

- Denominator \( = (x-3)(x+3) \).

Determine intervals:

- \( (x-3)(x+3) > 0 \) for \( x < -3 \) or \( x > 3 \).
- \( (x-3)(x+3) < 0 \) for \( -3 < x < 3 \).

Determine the sign of the numerator:

- Quadratic \( x^2 + 3x + 9 \) is always positive since it does not cross the x-axis (discriminant \( b^2-4ac = -27 \) is negative).

So \( A < 0 \) when \( (x-3)(x+3) < 0 \), i.e., for \( -3 < x < 3 \).

Thus, for \( A < 0 \), \( x \) must satisfy \( -3 < x < 3 \).