**Finding the Value of Alpha^2 and Beta^2**

To find the value of alpha^2 and beta^2, we need to first determine the values of alpha and beta. These values can be found by using the quadratic formula, which is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.

**Step 1: Finding the Values of Alpha and Beta**

Given that alpha and beta are the zeroes of the quadratic polynomial ax^2 + bx + c, we can set the equation equal to zero:

ax^2 + bx + c = 0

Using the quadratic formula, we can substitute the values of a, b, and c into the formula to find the values of alpha and beta:

alpha = (-b + √(b^2 - 4ac)) / 2a

beta = (-b - √(b^2 - 4ac)) / 2a

**Step 2: Finding the Values of Alpha^2 and Beta^2**

Now that we have the values of alpha and beta, we can find their squares:

alpha^2 = (-b + √(b^2 - 4ac))^2 / (2a)^2

beta^2 = (-b - √(b^2 - 4ac))^2 / (2a)^2

Simplifying these expressions further, we get:

alpha^2 = (b^2 - 2ab + (b^2 - 4ac)) / (4a^2)

beta^2 = (b^2 + 2ab + (b^2 - 4ac)) / (4a^2)

alpha^2 = (2b^2 - 4ab + 4ac) / (4a^2)

beta^2 = (2b^2 + 4ab + 4ac) / (4a^2)

Simplifying these expressions even further, we get:

alpha^2 = (b^2 - 2ab + 4ac) / (2a^2)

beta^2 = (b^2 + 2ab + 4ac) / (2a^2)

Therefore, the values of alpha^2 and beta^2 are given by the expressions:

alpha^2 = (b^2 - 2ab + 4ac) / (2a^2)

beta^2 = (b^2 + 2ab + 4ac) / (2a^2)

These expressions can be used to calculate the values of alpha^2 and beta^2, given the coefficients a, b, and c of the quadratic equation.