If (x+a) is a factor of `f(x)=(2x^2+2ax+5x+10)`, find a.

Options

A. 2
B. -1
C. 0
D. `(-3)/2`

Answer

To solve the problem, we need to determine the value of \( a \) such that \( (x + a) \) is a factor of the polynomial \( f(x) = 2x^2 + 2ax + 5x + 10 \).

### Step 1: Understand the Factor Condition
If \( (x + a) \) is a factor of \( f(x) \), then according to the factor theorem, \( f(-a) = 0 \).

### Step 2: Substitute \( -a \) into \( f(x) \)
We will substitute \( -a \) into the function \( f(x) \):
\[ f(-a) = 2(-a)^2 + 2a(-a) + 5(-a) + 10 \]

### Step 3: Simplify the Expression
Now, we simplify the expression:
\[ f(-a) = 2a^2 - 2a^2 - 5a + 10 \]
Notice that \( 2(-a)^2 = 2a^2 \) and \( 2a(-a) = -2a^2 \).

### Step 4: Combine Like Terms
Combining the terms gives us:
\[ f(-a) = 0 - 5a + 10 = -5a + 10 \]

### Step 5: Set the Equation to Zero
Since \( f(-a) = 0 \), we set the equation:
\[ -5a + 10 = 0 \]

### Step 6: Solve for \( a \)
Now, solve for \( a \):
\[ -5a = -10 \\ a = \frac{-10}{-5} = 2 \]

### Conclusion
Thus, the value of \( a \) is \( 2 \).