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Let's work through this partial fraction decomposition step-by-step, filling in the blanks to find \( A \) and \( B \).

The original equation is:
\[
\frac{16x + 34}{(2x + 2)(2x + 4)} = \frac{A}{2x + 2} + \frac{B}{2x + 4}
\]

To eliminate the denominators, we multiply each side by \((2x + 2)(2x + 4)\):
\[
16x + 34 = A(2x + 4) + B(2x + 2)
\]

Expanding \(A(2x + 4)\) and \(B(2x + 2)\):
- \(A(2x + 4) = 2Ax + 4A\)
- \(B(2x + 2) = 2Bx + 2B\)

So we get:
\[
16x + 34 = (2A + 2B)x + (4A + 2B)
\]

Now, we can match the coefficients of \(x\) and the constant terms on both sides:
1. For \(x\)-terms: \(2A + 2B = 16\)
2. For constants: \(4A + 2B = 34\)

Let's fill in the blanks based on this work.

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Would you like further details on the solution, or do you have any questions?

Here are some related questions for additional practice:
1. How would you solve for \(A\) and \(B\) if the constants on the right side were different?
2. What if the denominator had three linear factors? How would that change the setup?
3. How can you verify your values for \(A\) and \(B\) once solved?
4. What is the general approach to partial fraction decomposition when the denominator has repeated factors?
5. How would you set up a partial fraction decomposition if the numerator’s degree was equal to or greater than the denominator's?

**Tip:** Always expand and combine terms carefully in partial fraction decomposition to avoid mistakes in coefficients.

Solve for A and B by multiplying each side of the equation 16x + 34 / (2x + 2)(2x + 4) = A / (2x + 2) + B / (2x + 4) by the LCD and simplify.

Solve for A and B in the partial fraction decomposition of the rational expression 16x + 34 / (2x + 2)(2x + 4). Learn the steps to eliminate the denominators, expand, and combine terms. Suitable for students in Grades 10-12 studying algebra and partial fractions.

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