If `A = 4x ^(2) + 7x -1 and B =-x ^(2) -5x +3,` then what is `3/2 A -2B` ?

Options

A. `4x^(2) + (31)/(2)x -9/2`
B. `4x ^(2) + (41)/(2)x - (15)/(2)`
C. `8x ^(2) + (31)/(2)x -9/2`
D. `8x ^(2) + (41)/(2) x - (15)/(2)`

Answer

To solve the expression \( \frac{3}{2} A - 2B \) where \( A = 4x^2 + 7x - 1 \) and \( B = -x^2 - 5x + 3 \), we will follow these steps:

### Step 1: Substitute the values of A and B
We start by substituting the values of \( A \) and \( B \) into the expression \( \frac{3}{2} A - 2B \).

\[ \frac{3}{2} A - 2B = \frac{3}{2} (4x^2 + 7x - 1) - 2 (-x^2 - 5x + 3) \]

### Step 2: Distribute the constants
Next, we distribute \( \frac{3}{2} \) and \( -2 \) across the respective polynomials.

\[ = \frac{3}{2} \cdot 4x^2 + \frac{3}{2} \cdot 7x - \frac{3}{2} \cdot 1 + 2x^2 + 10x - 6 \]

Calculating each term gives:

\[ = 6x^2 + \frac{21}{2}x - \frac{3}{2} + 2x^2 + 10x - 6 \]

### Step 3: Combine like terms
Now, we will combine the like terms (terms with \( x^2 \), \( x \), and constant terms).

1. Combine \( x^2 \) terms:
\[ 6x^2 + 2x^2 = 8x^2 \]

2. Combine \( x \) terms:
\[ \frac{21}{2}x + 10x = \frac{21}{2}x + \frac{20}{2}x = \frac{41}{2}x \]

3. Combine constant terms:
\[ -\frac{3}{2} - 6 = -\frac{3}{2} - \frac{12}{2} = -\frac{15}{2} \]

### Step 4: Write the final expression
Putting it all together, we have:

\[ \frac{3}{2} A - 2B = 8x^2 + \frac{41}{2}x - \frac{15}{2} \]

### Final Answer
Thus, the final expression for \( \frac{3}{2} A - 2B \) is:

\[ 8x^2 + \frac{41}{2}x - \frac{15}{2} \]