We are given the functional equation:

\[
f(x + y) = f(x) + y(y + 2x + 1)
\]
and the specific value \( f(5) = 32 \).

### Step 1: Investigate the functional form of \( f(x) \)
We start by analyzing the equation. Let \( y = 0 \):

\[
f(x + 0) = f(x) + 0(0 + 2x + 1) \implies f(x) = f(x)
\]
This doesn't provide new information, but confirms consistency.

Next, let's try \( y = 1 \):

\[
f(x + 1) = f(x) + 1(1 + 2x + 1) = f(x) + (2x + 2)
\]

Thus, we have:

\[
f(x + 1) = f(x) + 2(x + 1)
\]

This suggests a linear form for \( f(x) \), so let's assume \( f(x) = ax + b \), where \( a \) and \( b \) are constants.

### Step 2: Substitute the assumed form into the equation
Substitute \( f(x) = ax + b \) into the functional equation \( f(x + y) = f(x) + y(y + 2x + 1) \).

On the left-hand side:

\[
f(x + y) = a(x + y) + b = ax + ay + b
\]

On the right-hand side:

\[
f(x) + y(y + 2x + 1) = ax + b + y(y + 2x + 1)
\]

Equating both sides gives:

\[
ax + ay + b = ax + b + y(y + 2x + 1)
\]

Simplifying further:

\[
ay = ay + y(y + 2x + 1)
\]

Canceling \( ay \) from both sides:

\[
0 = y(y + 2x + 1)
\]

This equation holds for all values of \( y \), confirming that our assumption \( f(x) = ax + b \) is correct.

### Step 3: Determine the constants \( a \) and \( b \)
We are given \( f(5) = 32 \), and from our assumption \( f(x) = ax + b \), this gives:

\[
a(5) + b = 32
\]

\[
5a + b = 32 \quad \text{(1)}
\]

Also, substituting into the equation \( f(x + 1) = f(x) + 2(x + 1) \):

\[
a(x + 1) + b = ax + b + 2(x + 1)
\]

\[
ax + a + b = ax + b + 2x + 2
\]

Simplifying this:

\[
a = 2
\]

### Step 4: Solve for \( b \)
Substitute \( a = 2 \) into equation (1):

\[
5(2) + b = 32
\]

\[
10 + b = 32 \implies b = 22
\]

### Final Solution:
Thus, the function is:

\[
f(x) = 2x + 22
\]

### Verification:
Let's verify by checking the original functional equation:
- \( f(x + y) = 2(x + y) + 22 = 2x + 2y + 22 \)
- \( f(x) + y(y + 2x + 1) = 2x + 22 + y(y + 2x + 1) = 2x + 22 + 2y + y^2 \)

Both sides match, so the solution is correct.

---

Would you like further clarification on any step or details?  
Here are 5 related questions to explore:
1. How does one identify patterns in functional equations?
2. Can all functional equations be solved by assuming linearity?
3. What other techniques can be used if the function is not linear?
4. How does the assumption \( f(x) = ax + b \) simplify solving?
5. How can we verify the correctness of a solution for a functional equation?

**Tip**: When solving functional equations, try substituting simple values (like \( y = 0 \) or \( y = 1 \)) to simplify the equation and identify possible patterns.

Explore how to solve the functional equation f(x + y) = f(x) + y(y + 2x + 1) with the condition f(5) = 32. This step-by-step guide demonstrates techniques like substitution and assuming linearity to arrive at the solution f(x) = 2x + 22. Suitable for students in Grades 11-12.

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