Determine the value of `k` for which `x+2` is a factor of `(x+1)^7+(2x+k)^3dot`

Explanation

To determine the value of \( k \) for which \( x + 2 \) is a factor of \( (x + 1)^7 + (2x + k)^3 \), we can follow these steps:

### Step 1: Substitute \( x = -2 \)
Since \( x + 2 \) is a factor, substituting \( x = -2 \) into the polynomial should yield 0.

\[ (-2 + 1)^7 + (2(-2) + k)^3 = 0 \]

### Step 2: Simplify the expression
Calculating the terms:

\[ (-1)^7 + ( -4 + k)^3 = 0 \]

This simplifies to:

\[ -1 + (k - 4)^3 = 0 \]

### Step 3: Rearranging the equation
Now, we can rearrange the equation:

\[ (k - 4)^3 = 1 \]

### Step 4: Taking the cube root
Taking the cube root of both sides gives:

\[ k - 4 = 1 \quad \text{or} \quad k - 4 = -1 \]

### Step 5: Solve for \( k \)
From \( k - 4 = 1 \):

\[ k = 5 \]

From \( k - 4 = -1 \):

\[ k = 3 \]

### Step 6: Check for real solutions
Now we need to check if both values of \( k \) yield real solutions. We will check the discriminant of the polynomial formed when substituting these values back into the original equation.

For \( k = 5 \):

\[ (k - 4)^3 = 1 \implies (5 - 4)^3 = 1 \implies 1 = 1 \quad \text{(valid)} \]

For \( k = 3 \):

\[ (k - 4)^3 = 1 \implies (3 - 4)^3 = -1 \quad \text{(not valid)} \]

Thus, the only valid solution is:

\[ \boxed{5} \]