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We are given the function \( p(x) = \frac{2x}{(x + 2)^2}, x \neq -2 \), and we are tasked to express it as a sum of two terms of the form \( \frac{A}{x + 2} + \frac{B}{(x + 2)^2} \), and then find the constants \( A \) and \( B \).

### Step 1: Set up the equation
We want to match the given expression with:

\[
p(x) = \frac{A}{x + 2} + \frac{B}{(x + 2)^2}.
\]

Start by rewriting the right-hand side with a common denominator:

\[
\frac{A}{x + 2} + \frac{B}{(x + 2)^2} = \frac{A(x + 2) + B}{(x + 2)^2}.
\]

Now, equate this to the given expression for \( p(x) \):

\[
\frac{A(x + 2) + B}{(x + 2)^2} = \frac{2x}{(x + 2)^2}.
\]

### Step 2: Compare numerators
Since the denominators are the same, we can equate the numerators:

\[
A(x + 2) + B = 2x.
\]

### Step 3: Expand and group terms
Now expand the left-hand side:

\[
A(x + 2) + B = Ax + 2A + B.
\]

This gives us:

\[
Ax + 2A + B = 2x.
\]

### Step 4: Compare coefficients
Now, compare the coefficients of \( x \) and the constant terms on both sides of the equation:

- Coefficient of \( x \): \( A = 2 \),
- Constant term: \( 2A + B = 0 \).

### Step 5: Solve for \( A \) and \( B \)
We already have \( A = 2 \). Substitute \( A = 2 \) into the second equation:

\[
2(2) + B = 0 \implies 4 + B = 0 \implies B = -4.
\]

### Final Answer:
The values of \( A \) and \( B \) are:

\[
A = 2, \quad B = -4.
\]

Would you like more details, or do you have any other questions?

### Related Questions:
1. How can we check if the values of \( A \) and \( B \) are correct?
2. What method can we use for partial fraction decomposition for higher-order polynomials?
3. Can this technique be used for improper rational functions?
4. How would you handle a situation where the denominator is not factored easily?
5. How does this process change if the power of \( (x + 2) \) in the denominator was higher than 2?

**Tip:** Always expand and simplify both sides of the equation when working with partial fractions to ensure the terms align correctly.

Find the values of the constants A and B such that p(x) = A / (x + 2) + B / (x + 2)^2, where p(x) = 2x / (x + 2)^2.

Learn how to find the constants A and B for the partial fraction decomposition of p(x) = 2x / (x + 2)^2. This problem involves algebraic techniques and partial fraction decomposition. Suitable for high school students in Grades 10-12.

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