To find the values of 'a' for which the equation has at least one real root, let's solve the equation step by step.

Step 1: Expand the equation
(x^2 + 2)^2 - (a-3)(x^2 + 1)(x^2 + 1)(a-4)(x^2 + 1)^2 = 0

Step 2: Simplify the equation
(x^4 + 4x^2 + 4) - (a-3)(x^2 + 1)(x^2 + 1)(a-4)(x^2 + 1)^2 = 0

Step 3: Expand and simplify further
x^4 + 4x^2 + 4 - (a-3)(x^2 + 1)(x^2 + 1)(a-4)(x^2 + 1)^2 = 0

Step 4: Expand and simplify the quadratic terms
x^4 + 4x^2 + 4 - (a-3)(x^2 + 1)(x^2 + 1)(a-4)(x^4 + 2x^2 + 1) = 0

Step 5: Distribute and combine like terms
x^4 + 4x^2 + 4 - (a-3)(a-4)(x^4 + 2x^2 + 1)(x^2 + 1) = 0

Step 6: Expand and simplify further
x^4 + 4x^2 + 4 - (a^2 - 7a + 12)(x^4 + 2x^2 + 1)(x^2 + 1) = 0

Step 7: Distribute and combine like terms
x^4 + 4x^2 + 4 - (a^2 - 7a + 12)(x^6 + 3x^4 + 3x^2 + x^2 + 1) = 0

Step 8: Simplify the equation
x^4 + 4x^2 + 4 - (a^2 - 7a + 12)(x^6 + 4x^4 + 4x^2 + 1) = 0

Step 9: Distribute and combine like terms
x^4 + 4x^2 + 4 - (a^2 - 7a + 12)x^6 - (a^2 - 7a + 12)4x^4 - (a^2 - 7a + 12)4x^2 - (a^2 - 7a + 12) = 0

Step 10: Rearrange the terms
(a^2 - 7a + 12)x^6 + (a^2 - 7a + 12)4x^4 + (a^2 - 7a + 12)4x^2 + x^4 + 4x^2 + 4 - (a^2 - 7a + 12) = 0

Step 11: Combine like terms
(a^2 - 7a + 12)x^6 + (4a^2 - 28a