`x^(2) + 4x + k` is divided by `(x-2)` leaves remainder 2x, find k

Answer

To solve the problem, we need to find the value of \( k \) such that when the polynomial \( f(x) = x^2 + 4x + k \) is divided by \( x - 2 \), the remainder is \( 2x \).

### Step-by-Step Solution:

1. **Understand the Remainder Theorem**: According to the Remainder Theorem, when a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). In this case, we are dividing by \( x - 2 \), so we will evaluate \( f(2) \).

**Hint**: Remember that the Remainder Theorem states that \( f(a) \) gives the remainder when \( f(x) \) is divided by \( x - a \).

2. **Set up the equation**: We know that the remainder when dividing \( f(x) \) by \( x - 2 \) is given as \( 2x \). Therefore, we need to find \( f(2) \) and set it equal to \( 2(2) \).

\[ f(2) = 2(2) = 4 \]

**Hint**: Substitute \( x = 2 \) into the expression for the remainder \( 2x \) to find the specific value.

3. **Calculate \( f(2) \)**: Now, substitute \( x = 2 \) into the polynomial \( f(x) = x^2 + 4x + k \):

\[ f(2) = 2^2 + 4(2) + k \]

Simplifying this gives:

\[ f(2) = 4 + 8 + k = 12 + k \]

**Hint**: Make sure to calculate each term carefully when substituting \( x = 2 \).

4. **Set the equation**: Now, we set \( f(2) \) equal to the remainder we calculated:

\[ 12 + k = 4 \]

**Hint**: This step involves setting up an equation based on the information given about the remainder.

5. **Solve for \( k \)**: To find \( k \), we can rearrange the equation:

\[ k = 4 - 12 \]
\[ k = -8 \]

**Hint**: Isolate \( k \) by moving other terms to the opposite side of the equation.

### Final Answer:
The value of \( k \) is \( -8 \).