If `x+2k` is a factor of `p(x)=x^5−4k^2x^3+2x+2k+3` . Find k.

Answer

To find the value of \( k \) such that \( x + 2k \) is a factor of the polynomial \( p(x) = x^5 - 4k^2x^3 + 2x + 2k + 3 \), we can follow these steps:

### Step 1: Set up the equation
Since \( x + 2k \) is a factor, we can substitute \( x = -2k \) into \( p(x) \) and set it equal to zero:
\[ p(-2k) = 0 \]

### Step 2: Substitute \( x = -2k \) into \( p(x) \)
Now we substitute \( -2k \) into the polynomial:
\[ p(-2k) = (-2k)^5 - 4k^2(-2k)^3 + 2(-2k) + 2k + 3 \]

### Step 3: Calculate each term
Calculating each term:
- The first term:
\[ (-2k)^5 = -32k^5 \]
- The second term:
\[ -4k^2(-2k)^3 = -4k^2(-8k^3) = 32k^5 \]
- The third term:
\[ 2(-2k) = -4k \]
- The fourth term:
\[ 2k \]
- The fifth term:
\[ 3 \]

### Step 4: Combine all terms
Now combine all these terms:
\[ p(-2k) = -32k^5 + 32k^5 - 4k + 2k + 3 \]
This simplifies to:
\[ p(-2k) = 0 - 2k + 3 \]

### Step 5: Set the equation to zero
Set the simplified expression equal to zero:
\[ -2k + 3 = 0 \]

### Step 6: Solve for \( k \)
Now solve for \( k \):
\[ -2k = -3 \implies k = \frac{3}{2} \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{\frac{3}{2}} \]