Resolve into partial fractions :
`(2x^(3)-3x^(2)-8x-26)/(2x^(2)-5x-12)`.

Answer

To resolve the expression \(\frac{2x^3 - 3x^2 - 8x - 26}{2x^2 - 5x - 12}\) into partial fractions, we will follow these steps:

### Step 1: Factor the Denominator
First, we need to factor the denominator \(2x^2 - 5x - 12\).

To factor \(2x^2 - 5x - 12\), we look for two numbers that multiply to \(2 \times -12 = -24\) and add to \(-5\). The numbers \(-8\) and \(3\) work because:
\[ -8 + 3 = -5 \quad \text{and} \quad -8 \times 3 = -24. \]
Thus, we can rewrite the expression as:
\[ 2x^2 - 8x + 3x - 12 = 2x(x - 4) + 3(x - 4) = (2x + 3)(x - 4). \]
So, the denominator factors to:
\[ 2x^2 - 5x - 12 = (2x + 3)(x - 4). \]

### Step 2: Perform Polynomial Long Division
Since the degree of the numerator \(2x^3 - 3x^2 - 8x - 26\) is greater than the degree of the denominator \(2x^2 - 5x - 12\), we perform polynomial long division.

1. Divide the leading term: \(\frac{2x^3}{2x^2} = x\).
2. Multiply the entire denominator by \(x\): \(x(2x + 3) + x(x - 4) = 2x^3 + 3x - 5x^2 - 12x\).
3. Subtract from the original numerator:
\[ (2x^3 - 3x^2 - 8x - 26) - (2x^3 - 5x^2 + 3x - 12) = 2x^2 - 5x - 14. \]
Thus, we have:
\[ \frac{2x^3 - 3x^2 - 8x - 26}{2x^2 - 5x - 12} = x + \frac{2x^2 - 5x - 14}{(2x + 3)(x - 4)}. \]

### Step 3: Set Up Partial Fractions
Now we need to resolve \(\frac{2x^2 - 5x - 14}{(2x + 3)(x - 4)}\) into partial fractions:
\[ \frac{2x^2 - 5x - 14}{(2x + 3)(x - 4)} = \frac{A}{2x + 3} + \frac{B}{x - 4}. \]

### Step 4: Clear the Denominator
Multiply through by the denominator \((2x + 3)(x - 4)\):
\[ 2x^2 - 5x - 14 = A(x - 4) + B(2x + 3). \]

### Step 5: Expand and Collect Like Terms
Expanding the right-hand side:
\[ 2x^2 - 5x - 14 = Ax - 4A + 2Bx + 3B = (A + 2B)x + (-4A + 3B). \]

### Step 6: Set Up the System of Equations
Now we can equate coefficients:
1. For \(x\): \(A + 2B = -5\).
2. For the constant term: \(-4A + 3B = -14\).

### Step 7: Solve the System of Equations
From the first equation:
\[ A = -5 - 2B. \]
Substituting into the second equation:
\[ -4(-5 - 2B) + 3B = -14 \implies 20 + 8B + 3B = -14 \implies 11B = -34 \implies B = -\frac{34}{11}. \]
Substituting \(B\) back to find \(A\):
\[ A = -5 - 2\left(-\frac{34}{11}\right) = -5 + \frac{68}{11} = \frac{-55 + 68}{11} = \frac{13}{11}. \]

### Step 8: Write the Partial Fraction Decomposition
Thus, we have:
\[ \frac{2x^2 - 5x - 14}{(2x + 3)(x - 4)} = \frac{13/11}{2x + 3} - \frac{34/11}{x - 4}. \]

### Final Result
Combining everything, we get:
\[ \frac{2x^3 - 3x^2 - 8x - 26}{2x^2 - 5x - 12} = x + \frac{13/11}{2x + 3} - \frac{34/11}{x - 4}. \]