Let's simplify the numerator of the expression for \( V_r - V_{r+2} \). Recall that the expression for \( V_r - V_{r+2} \) is:

\[
V_r - V_{r+2} = \frac{A(r + 1)(r + 2)(r + 3) - B(r)(r + 2)(r + 3) - A(r)(r + 1)(r + 3) + B(r)(r + 1)(r + 2)}{r(r + 1)(r + 2)(r + 3)}
\]

We need to simplify the numerator, which is:

\[
A(r + 1)(r + 2)(r + 3) - B(r)(r + 2)(r + 3) - A(r)(r + 1)(r + 3) + B(r)(r + 1)(r + 2)
\]

### Step 1: Expand Each Term
1. Expand \( A(r + 1)(r + 2)(r + 3) \):

\[
A(r + 1)(r + 2)(r + 3) = A[(r^2 + 3r + 2)(r + 3)] = A[r^3 + 6r^2 + 11r + 6]
\]

2. Expand \( B(r)(r + 2)(r + 3) \):

\[
B(r)(r + 2)(r + 3) = B[r(r^2 + 5r + 6)] = B[r^3 + 5r^2 + 6r]
\]

3. Expand \( A(r)(r + 1)(r + 3) \):

\[
A(r)(r + 1)(r + 3) = A[r(r^2 + 4r + 3)] = A[r^3 + 4r^2 + 3r]
\]

4. Expand \( B(r)(r + 1)(r + 2) \):

\[
B(r)(r + 1)(r + 2) = B[r(r^2 + 3r + 2)] = B[r^3 + 3r^2 + 2r]
\]

### Step 2: Combine the Expanded Terms
Now, substitute the expanded terms back into the numerator:

\[
A(r^3 + 6r^2 + 11r + 6) - B(r^3 + 5r^2 + 6r) - A(r^3 + 4r^2 + 3r) + B(r^3 + 3r^2 + 2r)
\]

Next, group like terms:

\[
= (Ar^3 - Br^3 - Ar^3 + Br^3) + (6Ar^2 - 5Br^2 - 4Ar^2 + 3Br^2) + (11Ar - 6Br - 3Ar + 2Br) + 6A
\]

### Step 3: Simplify the Grouped Terms
Simplify each group:

1. For the \( r^3 \) terms:
\[
(Ar^3 - Ar^3) + (Br^3 - Br^3) = 0
\]
2. For the \( r^2 \) terms:
\[
(6Ar^2 - 4Ar^2) + (-5Br^2 + 3Br^2) = 2Ar^2 - 2Br^2 = 2r^2(A - B)
\]
3. For the \( r \) terms:
\[
(11Ar - 3Ar) + (-6Br + 2Br) = 8Ar - 4Br = 4r(2A - B)
\]
4. Constant term:
\[
6A
\]

Thus, the simplified numerator is:
\[
2r^2(A - B) + 4r(2A - B) + 6A
\]

### Final Simplified Expression
The final expression for \( V_r - V_{r+2} \) is:
\[
V_r - V_{r+2} = \frac{2r^2(A - B) + 4r(2A - B) + 6A}{r(r + 1)(r + 2)(r + 3)}
\]

Now, to match this with \( U_r = \frac{2r + 3}{r(r + 1)(r + 2)(r + 3)} \), we will equate the coefficients from both numerators to solve for \( A \) and \( B \).

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Would you like to proceed with solving for \( A \) and \( B \) using the matching coefficients method, or do you have any other questions?

### Related Questions:
1. How do you expand products of polynomials efficiently?
2. What is the process for matching coefficients in algebraic expressions?
3. How do you factor polynomials from expanded forms?
4. Can you explain how common denominators simplify algebraic fractions?
5. What are common strategies for simplifying complex rational expressions?

### Tip:
When simplifying expressions, always look for common factors and group similar terms to make the process more manageable.

Learn how to simplify the numerator of a complex rational expression involving polynomial algebra. Step-by-step guide with detailed polynomial expansions and simplification strategies. Suitable for advanced high school and undergraduate students.

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