Given:
- Alpha and beta are roots of a quadratic equation
- Alpha * beta = 2
- Alpha^4 * beta^4 = 272

To find: The quadratic equation.

Method:
- We know that the quadratic equation will be of the form ax^2 + bx + c = 0, where a,b,c are constants.
- We also know that the roots of the quadratic equation are alpha and beta.
- Using the sum and product of roots of a quadratic equation, we can write the following equations:
- alpha + beta = -b/a
- alpha * beta = c/a
- We can substitute the given values of alpha * beta and alpha^4 * beta^4 in the second equation:
- alpha^2 * beta^2 = 4
- (alpha^2 * beta^2)^2 = 16
- alpha^4 * beta^4 = (alpha^2 * beta^2)^2 = 272
- We can simplify the above equations:
- alpha^2 * beta^2 = 4
- alpha^2 + 2alpha*beta + beta^2 = (alpha + beta)^2 = b^2/a^2
- We can substitute the value of alpha + beta in the above equation:
- (-b/a)^2 = (b^2/a^2) = 4 + 2*2 = 8
- b^2 = 8a^2
- We can substitute the values of alpha*beta and b^2 in the equation c/a:
- c/a = 2
- We can now write the quadratic equation in terms of a, b and c:
- ax^2 + bx + c = 0
- ax^2 + bx + 2a = 0 (substituting c/a = 2)
- x^2 + (b/a)x + 2 = 0 (dividing by a)
- x^2 + 2sqrt(2)x + 2 = 0 (substituting b^2 = 8a^2)

Answer:
The quadratic equation is x^2 + 2sqrt(2)x + 2 = 0.