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To resolve the given rational expression into partial fractions, we start by analyzing the expression:

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

Since the denominator consists of two linear factors, \((x+4)\) and \((x+5)\), we can express the fraction as the sum of two partial fractions:

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

### Step 1: Set up the equation
We multiply both sides of the equation by \((x+4)(x+5)\) to eliminate the denominators:

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

### Step 2: Expand the right-hand side
Now expand both terms on the right-hand side:

\[
A(x+5) = A x + 5A
\]
\[
B(x+4) = B x + 4B
\]

Thus, the equation becomes:

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

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

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

### Step 4: Solve the system of equations
We now solve this system of equations:

1. \(A + B = 3\)
2. \(5A + 4B = 5\)

Solve the first equation for \(A\):

\[
A = 3 - B
\]

Substitute this into the second equation:

\[
5(3 - B) + 4B = 5
\]
\[
15 - 5B + 4B = 5
\]
\[
15 - B = 5
\]
\[
B = 10
\]

Now substitute \(B = 10\) back into \(A + B = 3\):

\[
A + 10 = 3
\]
\[
A = -7
\]

### Step 5: Write the partial fractions
Thus, the partial fraction decomposition is:

\[
\frac{2x^2 + 3x + 5}{(x+4)(x+5)} = \frac{-7}{x+4} + \frac{10}{x+5}
\]

Would you like any further clarification or a more detailed explanation?

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### 5 Related Questions:
1. What are partial fractions used for in integration?
2. How do you decompose fractions with repeated linear factors?
3. What happens if the numerator has a higher degree than the denominator?
4. How would the method change for irreducible quadratic factors?
5. Can partial fractions be applied to complex numbers?

### Tip:
Always check if the numerator's degree is less than the denominator before applying partial fractions. If not, perform polynomial long division first.

Resolve in partial fractions: \(\frac{2x^2 + 3x + 5}{(x+4)(x+5)}\)

Learn how to decompose the rational expression (2x^2 + 3x + 5) / ((x+4)(x+5)) into partial fractions with detailed steps. This algebra problem involves setting up and solving a system of linear equations, suitable for students in Grades 11-12.

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