The problem asks us to find the set of values for k for which the given algebraic expression results in an integer.

The expression is given as:

$$ E(k) = \frac{{{{\left( {k + 2} \right)}^2}}}{{k - 3}} $$

To determine when this expression yields an integer, we can simplify it. Let's use a substitution to make the expression easier to handle. Let \( x = k - 3 \). This implies that \( k = x + 3 \).

Now, substitute \( k = x + 3 \) into the expression \( k + 2 \):

$$ k + 2 = (x + 3) + 2 = x + 5 $$

Substitute these back into the original expression \( E(k) \):

$$ E(k) = \frac{{(x + 5)^2}}{x} $$

Expand the numerator:

$$ E(k) = \frac{{x^2 + 2 \cdot x \cdot 5 + 5^2}}{x} = \frac{{x^2 + 10x + 25}}{x} $$

Now, divide each term in the numerator by \( x \):

$$ E(k) = \frac{{x^2}}{x} + \frac{{10x}}{x} + \frac{{25}}{x} = x + 10 + \frac{{25}}{x} $$

For \( E(k) \) to be an integer, the term \( \frac{{25}}{x} \) must also be an integer, since \( x \) (which is \( k - 3 \)) and \( 10 \) are integers (assuming \( k \) is an integer). This means that \( x \) must be a divisor of 25.

Finding Possible Values for x

The divisors of 25 are the integers that divide 25 without leaving a remainder. These are:

\( \pm 1, \pm 5, \pm 25 \)

Determining Values for k

Now, we need to find the corresponding values of \( k \) using the relationship \( k = x + 3 \):

• If \( x = 1 \), then \( k = 1 + 3 = 4 \).
• If \( x = -1 \), then \( k = -1 + 3 = 2 \).
• If \( x = 5 \), then \( k = 5 + 3 = 8 \).
• If \( x = -5 \), then \( k = -5 + 3 = -2 \).
• If \( x = 25 \), then \( k = 25 + 3 = 28 \).
• If \( x = -25 \), then \( k = -25 + 3 = -22 \).

So, the possible integer values for \( k \) that make the expression an integer are \( \{4, 2, 8, -2, 28, -22\} \).

Checking the Options

We need to find which of the given options contains only values from this set.

Let's verify the values in Option 3:

• For \( k = 4 \): \( \frac{{(4 + 2)^2}}{{4 - 3}} = \frac{{6^2}}{1} = 36 \), which is an integer.
• For \( k = 8 \): \( \frac{{(8 + 2)^2}}{{8 - 3}} = \frac{{10^2}}{5} = \frac{100}{5} = 20 \), which is an integer.
• For \( k = 28 \): \( \frac{{(28 + 2)^2}}{{28 - 3}} = \frac{{30^2}}{25} = \frac{900}{25} = 36 \), which is an integer.

All values in Option 3 satisfy the condition.