To solve this problem, we need to set up a system of equations based on the given information and then solve for the values of the two numbers.

Let's assume the two numbers are x and y.

1. Setting up the equations:
According to the given information, the difference between the two numbers is 9, so we can write the first equation as:
x - y = 9

The product of the two numbers is 14, so the second equation can be written as:
xy = 14

2. Solving the equations:
We can solve the system of equations by substitution or elimination method. Let's use the substitution method in this case.

From the first equation, we can rewrite it as:
x = y + 9

Substituting this value of x into the second equation, we get:
(y + 9)y = 14

Expanding and rearranging the equation, we get:
y^2 + 9y - 14 = 0

Now we can solve this quadratic equation by factoring or using the quadratic formula. In this case, the equation factors as:
(y + 14)(y - 1) = 0

Setting each factor equal to zero, we get two possible solutions:
y + 14 = 0 --> y = -14
y - 1 = 0 --> y = 1

So the two possible values for y are -14 and 1.

3. Finding the sum of the numbers and its square:
Now that we have the values of y, we can find the corresponding values of x by substituting them back into the equation x = y + 9.

For y = -14:
x = -14 + 9
x = -5

For y = 1:
x = 1 + 9
x = 10

The sum of the two numbers is -5 + 1 = -4. And the square of -4 is 16.

Therefore, the correct answer is option C) 137.