If `(x+5)(x+p)=x^(2)+2x+k`, then

Options

A. p=3 and k=5
B. p=-3 and k=15
C. p=3 and k=-15
D. p=-3 and k=-15

Answer

To solve the equation \((x+5)(x+p) = x^2 + 2x + k\), we will follow these steps:

### Step 1: Expand the left-hand side
We start by expanding the left-hand side of the equation:
\[ (x + 5)(x + p) = x^2 + px + 5x + 5p \]
Combining like terms, we get:
\[ x^2 + (p + 5)x + 5p \]

### Step 2: Set the expanded form equal to the right-hand side
Now, we set the expanded form equal to the right-hand side of the equation:
\[ x^2 + (p + 5)x + 5p = x^2 + 2x + k \]

### Step 3: Compare coefficients
Since the expressions are equal, we can compare the coefficients of \(x\) and the constant terms on both sides.

1. For the coefficient of \(x\):
\[ p + 5 = 2 \]
Solving for \(p\):
\[ p = 2 - 5 = -3 \]

2. For the constant term:
\[ 5p = k \]
Substituting \(p = -3\):
\[ k = 5(-3) = -15 \]

### Step 4: State the final values
Thus, we find:
\[ p = -3 \quad \text{and} \quad k = -15 \]

### Final Answer
The values are:
\[ p = -3 \quad \text{and} \quad k = -15 \]

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