In the product of `(9x ^(2)+ 15 -x) and (-1 -x + x ^(2)),` if A, B and C are the coefficients of `x ^(3), x ^(2) and x` respectively, then the value of (A+B-C) is

Options

A. `-3`
B. 14
C. `-17`
D. 11

Answer

To solve the problem, we need to find the product of the two polynomials \( (9x^2 + 15 - x) \) and \( (-1 - x + x^2) \), and then determine the coefficients \( A \), \( B \), and \( C \) corresponding to \( x^3 \), \( x^2 \), and \( x \) respectively. Finally, we will compute \( A + B - C \).

### Step-by-step Solution:

1. **Write the Polynomials**:
\[ P(x) = 9x^2 + 15 - x \]
\[ Q(x) = -1 - x + x^2 \]

2. **Multiply the Polynomials**:
We will multiply \( P(x) \) and \( Q(x) \) using the distributive property (also known as the FOIL method for binomials).

\[ P(x) \cdot Q(x) = (9x^2)(-1) + (9x^2)(-x) + (9x^2)(x^2) + (15)(-1) + (15)(-x) + (15)(x^2) + (-x)(-1) + (-x)(-x) + (-x)(x^2) \]

Now, let's calculate each term:
- \( 9x^2 \cdot -1 = -9x^2 \)
- \( 9x^2 \cdot -x = -9x^3 \)
- \( 9x^2 \cdot x^2 = 9x^4 \)
- \( 15 \cdot -1 = -15 \)
- \( 15 \cdot -x = -15x \)
- \( 15 \cdot x^2 = 15x^2 \)
- \( -x \cdot -1 = x \)
- \( -x \cdot -x = x^2 \)
- \( -x \cdot x^2 = -x^3 \)

3. **Combine Like Terms**:
Now we will combine all the terms we calculated:
\[ 9x^4 + (-9x^3 - x^3) + (-9x^2 + 15x^2) + (-15 - 15x + x) \]
This simplifies to:
\[ 9x^4 - 10x^3 + 7x^2 - 14x - 15 \]

4. **Identify Coefficients**:
From the combined polynomial:
- Coefficient of \( x^3 \) (A) = \( -10 \)
- Coefficient of \( x^2 \) (B) = \( 7 \)
- Coefficient of \( x \) (C) = \( -14 \)

5. **Calculate \( A + B - C \)**:
Now we calculate:
\[ A + B - C = -10 + 7 - (-14) \]
Simplifying this:
\[ = -10 + 7 + 14 = 11 \]

### Final Answer:
The value of \( A + B - C \) is \( 11 \).