Resolve into Partial Fractions
(v) `(13x+43)/(2x^(2)+17x+30)`

Answer

To resolve the expression \(\frac{13x + 43}{2x^2 + 17x + 30}\) into partial fractions, we will follow these steps:

### Step 1: Factor the Denominator
First, we need to factor the quadratic expression in the denominator, \(2x^2 + 17x + 30\).

To factor \(2x^2 + 17x + 30\), we look for two numbers that multiply to \(2 \times 30 = 60\) and add up to \(17\). The numbers \(12\) and \(5\) work since \(12 \times 5 = 60\) and \(12 + 5 = 17\).

So we can rewrite the quadratic as:
\[ 2x^2 + 12x + 5x + 30 \]
Now, we can group the terms:
\[ = 2x(x + 6) + 5(x + 6) = (2x + 5)(x + 6) \]

### Step 2: Set Up the Partial Fraction Decomposition
Now that we have factored the denominator, we can express the fraction as:
\[ \frac{13x + 43}{(2x + 5)(x + 6)} = \frac{A}{x + 6} + \frac{B}{2x + 5} \]
where \(A\) and \(B\) are constants we need to determine.

### Step 3: Combine the Right Side
To combine the right side into a single fraction, we have:
\[ \frac{A(2x + 5) + B(x + 6)}{(2x + 5)(x + 6)} = \frac{13x + 43}{(2x + 5)(x + 6)} \]
This gives us the equation:
\[ 13x + 43 = A(2x + 5) + B(x + 6) \]

### Step 4: Expand and Collect Like Terms
Expanding the right side:
\[ A(2x + 5) + B(x + 6) = 2Ax + 5A + Bx + 6B = (2A + B)x + (5A + 6B) \]
So we have:
\[ 13x + 43 = (2A + B)x + (5A + 6B) \]

### Step 5: Set Up the System of Equations
Now, we can equate the coefficients of \(x\) and the constant terms:
1. \(2A + B = 13\) (coefficient of \(x\))
2. \(5A + 6B = 43\) (constant term)

### Step 6: Solve the System of Equations
From the first equation, we can express \(B\) in terms of \(A\):
\[ B = 13 - 2A \]
Substituting this into the second equation:
\[ 5A + 6(13 - 2A) = 43 \]
Expanding gives:
\[ 5A + 78 - 12A = 43 \]
Combining like terms:
\[ -7A + 78 = 43 \]
Subtracting \(78\) from both sides:
\[ -7A = 43 - 78 \]
\[ -7A = -35 \]
Dividing by \(-7\):
\[ A = 5 \]

Now substituting \(A = 5\) back into the equation for \(B\):
\[ B = 13 - 2(5) = 13 - 10 = 3 \]

### Step 7: Write the Final Partial Fraction Decomposition
Now that we have \(A\) and \(B\):
\[ \frac{13x + 43}{2x^2 + 17x + 30} = \frac{5}{x + 6} + \frac{3}{2x + 5} \]

### Final Answer
Thus, the partial fraction decomposition is:
\[ \frac{13x + 43}{2x^2 + 17x + 30} = \frac{5}{x + 6} + \frac{3}{2x + 5} \]