If α and β are zeroes of the polynomial `P(x) = 2x^(2) + 11x + 5`, find the value of `(1)/(alpha) + (1)/(beta) - 2 alpha beta`

Answer

To solve the problem, we need to find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} - 2 \alpha \beta \) where \( \alpha \) and \( \beta \) are the roots of the polynomial \( P(x) = 2x^2 + 11x + 5 \).

### Step 1: Identify coefficients
The polynomial is in the form \( ax^2 + bx + c \), where:
- \( a = 2 \)
- \( b = 11 \)
- \( c = 5 \)

### Step 2: Use Vieta's formulas
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{11}{2} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{5}{2} \)

### Step 3: Substitute values into the expression
Now, we substitute these values into the expression \( \frac{1}{\alpha} + \frac{1}{\beta} - 2 \alpha \beta \).

First, we simplify \( \frac{1}{\alpha} + \frac{1}{\beta} \):
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]
Substituting the values we found:
\[ \frac{\beta + \alpha}{\alpha \beta} = \frac{-\frac{11}{2}}{\frac{5}{2}} = \frac{-11}{5} \]

Next, we calculate \( -2 \alpha \beta \):
\[ -2 \alpha \beta = -2 \times \frac{5}{2} = -5 \]

### Step 4: Combine the results
Now we combine the results:
\[ \frac{1}{\alpha} + \frac{1}{\beta} - 2 \alpha \beta = \frac{-11}{5} - 5 \]
To combine these, we need a common denominator:
\[ -5 = \frac{-25}{5} \]
So,
\[ \frac{-11}{5} - \frac{25}{5} = \frac{-11 - 25}{5} = \frac{-36}{5} \]

### Final Answer
Thus, the value of \( \frac{1}{\alpha} + \frac{1}{\beta} - 2 \alpha \beta \) is:
\[ \frac{-36}{5} \]