Consider the equation ` x ^(3) -ax ^(2) +bx-c=0,` where a,b,c are rational number, `a ne 1.` it is given that ` x _(1), x _(2) and x _(1)x_(2)` are the real roots of the equation. Then `x _(1) x _(2) ((a +1)/(b +c))=`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to analyze the given cubic equation and its roots. The equation is:

\[ x^3 - ax^2 + bx - c = 0 \]

where \( a, b, c \) are rational numbers and \( a \neq 1 \). The roots of the equation are given as \( x_1, x_2, \) and \( x_1 x_2 \).

### Step 1: Sum of the Roots
According to Vieta's formulas, the sum of the roots of a cubic equation \( x^3 + px^2 + qx + r = 0 \) is given by:

\[ x_1 + x_2 + x_1 x_2 = a \]

### Step 2: Sum of the Products of the Roots Taken Two at a Time
The sum of the products of the roots taken two at a time is given by:

\[ x_1 x_2 + x_1 (x_1 x_2) + x_2 (x_1 x_2) = b \]

This can be simplified as:

\[ x_1 x_2 + x_1^2 x_2 + x_2^2 x_1 = b \]

### Step 3: Product of the Roots
The product of the roots taken three at a time is given by:

\[ x_1 x_2 (x_1 x_2) = c \]

This simplifies to:

\[ (x_1 x_2)^2 = c \]

### Step 4: Finding the Expression
We need to find the value of the expression:

\[ x_1 x_2 \left( \frac{a + 1}{b + c} \right) \]

### Step 5: Substitute Values
From our earlier steps, we can substitute the values of \( a, b, \) and \( c \):

1. \( a = x_1 + x_2 + x_1 x_2 \)
2. \( b = x_1 x_2 + x_1^2 x_2 + x_2^2 x_1 \)
3. \( c = (x_1 x_2)^2 \)

Now substituting these into the expression:

\[ x_1 x_2 \left( \frac{(x_1 + x_2 + x_1 x_2) + 1}{(x_1 x_2 + x_1^2 x_2 + x_2^2 x_1) + (x_1 x_2)^2} \right) \]

### Step 6: Simplifying the Denominator
The denominator can be simplified as:

\[ b + c = (x_1 x_2 + x_1^2 x_2 + x_2^2 x_1) + (x_1 x_2)^2 \]

### Step 7: Factorization
Now, we can factor out \( x_1 x_2 \) from the numerator and denominator:

\[ x_1 x_2 \left( \frac{x_1 + x_2 + x_1 x_2 + 1}{x_1 x_2 + x_1^2 x_2 + x_2^2 x_1 + (x_1 x_2)^2} \right) \]

### Step 8: Final Calculation
After simplification, we find that:

\[ \frac{x_1 + x_2 + x_1 x_2 + 1}{x_1 x_2 + x_1^2 x_2 + x_2^2 x_1 + (x_1 x_2)^2} = 1 \]

Thus, the final value of the expression is:

\[ x_1 x_2 \cdot 1 = x_1 x_2 \]

### Conclusion
The final answer is:

\[ \boxed{1} \]