Perform the given operations.$\left(a^2+4ac+4c^2\right)\left(a^2-4c^2\right)$

Step by Step Solution: Step 1. We can use the formula (a+b)(a-b) = a^2 - b^2 to simplify the given expression Step 2. (a^2 + 4ac + 4c^2)(a^2 - 4c^2) = [(a+2c)(a+2c)][(a+2c)(a-2c)] Step 3. Using the formula (a+b)(a+b) = (a+b)^2 = a^2 + 2ab + b^2, Step 4. [(a+2c)(a+2c)][(a+2c)(a-2c)] = [(a+2c)^2][(a-2c)^2] = [(a^2 + 4ac + 4c^2)][(a^2 - 4ac + 4c^2)] Step 5. Multiplying the expressions within the brackets, Step 6. [(a^2 + 4ac + 4c^2)][(a^2 - 4ac + 4c^2)] = (a^2)^2 - (4ac)^2 Step 7. Substituting the values of a and c, Step 8. (a^2)^2 - (4ac)^2 = a^4 - 16c^2a^2 Final Answer: a^4 - 16c^2a^2