If a and b are the roots of `x^2 - 2x - 1 = 0`, then value of `a^2b + ab^2` is:

Options

A. `-2`
B. 2
C. `1/2`
D. 4

Answer

To solve the problem, we need to find the value of \( a^2b + ab^2 \) where \( a \) and \( b \) are the roots of the equation \( x^2 - 2x - 1 = 0 \).

### Step 1: Identify the coefficients of the quadratic equation
The given quadratic equation is:
\[ x^2 - 2x - 1 = 0 \]
Here, the coefficients are:
- \( A = 1 \)
- \( B = -2 \)
- \( C = -1 \)

### Step 2: Use Vieta's formulas to find \( a + b \) and \( ab \)
According to Vieta's formulas:
- The sum of the roots \( a + b = -\frac{B}{A} = -\frac{-2}{1} = 2 \)
- The product of the roots \( ab = \frac{C}{A} = \frac{-1}{1} = -1 \)

### Step 3: Rewrite the expression \( a^2b + ab^2 \)
We can factor the expression \( a^2b + ab^2 \):
\[ a^2b + ab^2 = ab(a + b) \]

### Step 4: Substitute the values of \( ab \) and \( a + b \)
Now we substitute the values we found:
- \( ab = -1 \)
- \( a + b = 2 \)

Thus, we have:
\[ a^2b + ab^2 = ab(a + b) = (-1)(2) = -2 \]

### Final Answer
The value of \( a^2b + ab^2 \) is:
\[ \boxed{-2} \]