If `p=x(3x+5)-28`, then p is divisible by which of the following expression?

Options

A. `3x+4`
B. `x-4`
C. `x+7`
D. `3x-7`

Answer

To solve the problem, we need to factor the expression \( p = x(3x + 5) - 28 \) and determine which expressions are factors of \( p \).

### Step 1: Expand the expression
First, we need to expand the expression for \( p \):
\[ p = x(3x + 5) - 28 \]
Expanding this gives:
\[ p = 3x^2 + 5x - 28 \]

### Step 2: Identify coefficients
Now, we identify the coefficients of the quadratic polynomial \( 3x^2 + 5x - 28 \):
- \( a = 3 \)
- \( b = 5 \)
- \( c = -28 \)

### Step 3: Use the splitting the middle term method
We need to find two numbers \( p \) and \( q \) such that:
- \( p + q = b = 5 \)
- \( p \cdot q = a \cdot c = 3 \cdot (-28) = -84 \)

### Step 4: Find suitable \( p \) and \( q \)
We need to find two numbers that multiply to \(-84\) and add up to \(5\). After testing possible pairs, we find:
- \( p = 12 \)
- \( q = -7 \)

### Step 5: Rewrite the middle term
Now we rewrite the polynomial using \( p \) and \( q \):
\[ p = 3x^2 + 12x - 7x - 28 \]

### Step 6: Group the terms
Next, we group the terms:
\[ p = (3x^2 + 12x) + (-7x - 28) \]

### Step 7: Factor by grouping
Now, we factor out the common factors from each group:
\[ p = 3x(x + 4) - 7(x + 4) \]
Now, we can factor out \( (x + 4) \):
\[ p = (x + 4)(3x - 7) \]

### Step 8: Identify the factors
From the factorization, we see that \( p \) can be expressed as:
\[ p = (x + 4)(3x - 7) \]
Thus, the factors of \( p \) are \( (x + 4) \) and \( (3x - 7) \).

### Conclusion
Therefore, \( p \) is divisible by both \( (x + 4) \) and \( (3x - 7) \). If the options provided include \( 3x - 7 \), then that is the correct answer.