In the product of `(5x +2) and (2x ^(2) - 3x +5), ` the sum of the coefficients of `x ^(2) and x ` is

Options

A. 9
B. `-8`
C. `-9`
D. 8

Answer

To solve the problem, we need to find the sum of the coefficients of \(x^2\) and \(x\) in the product of the expressions \((5x + 2)\) and \((2x^2 - 3x + 5)\).

### Step-by-Step Solution:

1. **Write down the expressions**:
We have:
\[ (5x + 2) \quad \text{and} \quad (2x^2 - 3x + 5) \]

2. **Multiply the two expressions**:
We will use the distributive property (also known as the FOIL method for binomials) to multiply the two expressions:
\[ (5x + 2)(2x^2 - 3x + 5) \]

3. **Distribute \(5x\)**:
- \(5x \cdot 2x^2 = 10x^3\)
- \(5x \cdot (-3x) = -15x^2\)
- \(5x \cdot 5 = 25x\)

4. **Distribute \(2\)**:
- \(2 \cdot 2x^2 = 4x^2\)
- \(2 \cdot (-3x) = -6x\)
- \(2 \cdot 5 = 10\)

5. **Combine all the terms**:
Now, we combine all the terms we obtained:
\[ 10x^3 + (-15x^2 + 4x^2) + (25x - 6x) + 10 \]
This simplifies to:
\[ 10x^3 - 11x^2 + 19x + 10 \]

6. **Identify the coefficients**:
- The coefficient of \(x^2\) is \(-11\).
- The coefficient of \(x\) is \(19\).

7. **Sum the coefficients**:
Now, we find the sum of the coefficients of \(x^2\) and \(x\):
\[ -11 + 19 = 8 \]

### Final Answer:
The sum of the coefficients of \(x^2\) and \(x\) is \(8\).

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