If `[(3a + 5b) (c) (d)] = p [acd] + q[bcd]`, them `p + q=`

Options

A. 8
B. `-8`
C. 2
D. 0

Answer

To solve the problem, we need to analyze the given expression and relate it to the box product (scalar triple product) of vectors.

### Step-by-Step Solution:

1. **Understanding the Given Expression**:
We have the expression:
\[ [(3a + 5b) \cdot (c \times d)] = p [a \cdot (c \times d)] + q [b \cdot (c \times d)] \]
Here, \( (3a + 5b) \) is a vector and \( (c \times d) \) is another vector.

2. **Using the Properties of Dot Product**:
The dot product distributes over addition, so we can rewrite the left-hand side:
\[ (3a + 5b) \cdot (c \times d) = 3(a \cdot (c \times d)) + 5(b \cdot (c \times d)) \]

3. **Identifying the Box Product**:
The expressions \( a \cdot (c \times d) \) and \( b \cdot (c \times d) \) can be represented as box products:
\[ a \cdot (c \times d) = [a, c, d] \quad \text{and} \quad b \cdot (c \times d) = [b, c, d] \]

4. **Rewriting the Equation**:
Now we can rewrite the equation as:
\[ 3[a, c, d] + 5[b, c, d] = p[a, c, d] + q[b, c, d] \]

5. **Comparing Coefficients**:
To find the values of \( p \) and \( q \), we can compare coefficients from both sides of the equation:
- Coefficient of \( [a, c, d] \): \( p = 3 \)
- Coefficient of \( [b, c, d] \): \( q = 5 \)

6. **Calculating \( p + q \)**:
Now we can find \( p + q \):
\[ p + q = 3 + 5 = 8 \]

### Final Answer:
Thus, the value of \( p + q \) is:
\[ \boxed{8} \]