**Problem:**
Find the value of 'k' in the polynomial equation x^2 + 6x + k = 0, given that the difference between its zeroes is 2.

**Solution:**

We are given a quadratic polynomial equation x^2 + 6x + k = 0 and we need to find the value of 'k' when the difference between its zeroes is 2.

Let's solve this problem step by step:

**Step 1: Understand the question**
We need to find the value of 'k' when the difference between the zeroes of the given quadratic equation is 2.

**Step 2: Recall the concept of zeroes of a quadratic equation**
The zeroes of a quadratic equation ax^2 + bx + c = 0 are the values of 'x' at which the equation equals zero. In other words, they are the values of 'x' that satisfy the equation.

**Step 3: Understand the relationship between the zeroes and coefficients**
For a quadratic equation ax^2 + bx + c = 0, the sum of the zeroes is given by the formula:
Sum of zeroes (α + β) = -b/a
The product of the zeroes is given by the formula:
Product of zeroes (α * β) = c/a

**Step 4: Use the given information to form equations**
We are given that the difference between the zeroes (α - β) is 2. Using this information, we can form the equation:
α - β = 2

**Step 5: Form equations using the relationship between zeroes and coefficients**
We can use the relationship between the zeroes and coefficients to form two more equations.
From the given equation, we can identify that a = 1, b = 6, and c = k.
Using the formulas mentioned in Step 3, we can form the following equations:
α + β = -b/a = -6/1 = -6
α * β = c/a = k/1 = k

**Step 6: Solve the equations**
We now have a system of three equations:
α - β = 2
α + β = -6
α * β = k

We can solve this system of equations using various methods like substitution, elimination, or graphing. Let's solve it using the elimination method.

Adding the first two equations, we get:
(α - β) + (α + β) = 2 + (-6)
2α = -4
α = -2

Now, substituting the value of α in the second equation, we get:
-2 + β = -6
β = -6 + 2
β = -4

**Step 7: Find the value of 'k'**
Finally, we need to find the value of 'k'. We can substitute the values of α and β in the equation α * β = k to get:
-2 * (-4) = k
8 = k

Therefore, the value of 'k' is 8.

**Step 8: Verify the solution**
To verify our solution, we can substitute the values of α, β, and k in the given quadratic equation and check if it satisfies the equation.
Substituting α = -2, β = -4, and k = 8 in the equation x^2 + 6x + k = 0, we get: