1. The problem is to factor completely the expression $$48x^{2n} - 75x^{4n}$$. 2. First, identify the greatest common factor (GCF) of the terms. Both terms have a factor of $$3x^{2n}$$. 3. Factor out the GCF: $$48x^{2n} - 75x^{4n} = 3x^{2n}(16 - 25x^{2n})$$. 4. Notice that inside the parentheses, $$16 - 25x^{2n}$$ is a difference of squares because: $$16 = 4^2$$ and $$25x^{2n} = (5x^n)^2$$. 5. Apply the difference of squares formula: $$a^2 - b^2 = (a - b)(a + b)$$. 6. So, $$16 - 25x^{2n} = (4 - 5x^n)(4 + 5x^n)$$. 7. Therefore, the complete factorization is: $$48x^{2n} - 75x^{4n} = 3x^{2n}(4 - 5x^n)(4 + 5x^n)$$. This is the fully factored form.