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The expression you've provided is:

\[
\frac{x^2 + 3x + 4}{(2x + 2)(2x^2 + 5)}
\]

To resolve this expression into partial fractions, we will express it in the form:

\[
\frac{x^2 + 3x + 4}{(2x + 2)(2x^2 + 5)} = \frac{A}{2x + 2} + \frac{Bx + C}{2x^2 + 5}
\]

Here, \( A \), \( B \), and \( C \) are constants that we need to determine.

### Step 1: Clear the denominator
Multiply both sides by the denominator \((2x + 2)(2x^2 + 5)\):

\[
x^2 + 3x + 4 = A(2x^2 + 5) + (Bx + C)(2x + 2)
\]

### Step 2: Expand and collect like terms
Expand the right side:

\[
x^2 + 3x + 4 = A(2x^2 + 5) + (Bx + C)(2x + 2)
\]

Expanding further:

\[
x^2 + 3x + 4 = A \cdot 2x^2 + 5A + Bx \cdot 2x + Bx \cdot 2 + C \cdot 2x + C \cdot 2
\]

\[
x^2 + 3x + 4 = 2Ax^2 + 5A + 2Bx^2 + 2Bx + 2Cx + 2C
\]

Combine like terms:

\[
x^2 + 3x + 4 = (2A + 2B)x^2 + (2B + 2C)x + (5A + 2C)
\]

### Step 3: Equate coefficients
Now, equate the coefficients of corresponding powers of \(x\) on both sides:

- Coefficient of \(x^2\): \(2A + 2B = 1\)
- Coefficient of \(x\): \(2B + 2C = 3\)
- Constant term: \(5A + 2C = 4\)

### Step 4: Solve the system of equations
Let's solve these equations:

1. \(2A + 2B = 1 \Rightarrow A + B = \frac{1}{2}\)
2. \(2B + 2C = 3 \Rightarrow B + C = \frac{3}{2}\)
3. \(5A + 2C = 4\)

From the first equation:

\[
B = \frac{1}{2} - A
\]

Substitute \(B\) in the second equation:

\[
\frac{1}{2} - A + C = \frac{3}{2} \Rightarrow C = A + 1
\]

Now, substitute \(C\) into the third equation:

\[
5A + 2(A + 1) = 4 \Rightarrow 7A + 2 = 4 \Rightarrow 7A = 2 \Rightarrow A = \frac{2}{7}
\]

Substitute \(A\) into the expressions for \(B\) and \(C\):

\[
B = \frac{1}{2} - \frac{2}{7} = \frac{7}{14} - \frac{4}{14} = \frac{3}{14}
\]

\[
C = \frac{2}{7} + 1 = \frac{2}{7} + \frac{7}{7} = \frac{9}{7}
\]

### Final Answer
So, the partial fraction decomposition of the given expression is:

\[
\frac{x^2 + 3x + 4}{(2x + 2)(2x^2 + 5)} = \frac{\frac{2}{7}}{2x + 2} + \frac{\frac{3}{14}x + \frac{9}{7}}{2x^2 + 5}
\]

Would you like more details or have any questions?

---

Here are five related questions to consider:

1. How would the partial fraction decomposition change if the numerator was \(x^2 + 3x + 1\) instead of \(x^2 + 3x + 4\)?
2. Can you solve the partial fraction decomposition if the denominator were \( (2x + 1)(x^2 + 2) \)?
3. What are the implications if the quadratic in the numerator is factorable?
4. How do the constants \( A \), \( B \), and \( C \) affect the shape of the graph of the partial fractions?
5. How can you confirm that the partial fraction decomposition is correct?

**Tip:** Always double-check your expanded expressions and coefficients before solving the system of equations in partial fractions.

Learn how to resolve the expression x^2 + 3x + 4 / (2x + 2)(2x^2 + 5) into partial fractions. Step-by-step solution including clearing the denominator, expanding and collecting like terms, equating coefficients, and solving the system of equations. Suitable for advanced high school students studying algebra and partial fraction decomposition.

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