1. Stating the problem: We want to simplify the expression $(4+3c^5)(4-3c^5)$.\n\n2. This expression is in the form of $(a+b)(a-b)$, which is a difference of squares formula: $$ (a+b)(a-b) = a^2 - b^2 $$\n\n3. Identify $a=4$ and $b=3c^5$. Substitute into the formula: $$ (4+3c^5)(4-3c^5) = 4^2 - (3c^5)^2 $$\n\n4. Calculate each square: $$ 4^2 = 16 $$ and $$ (3c^5)^2 = 3^2 \times (c^5)^2 = 9c^{10} $$\n\n5. Substitute these back: $$ 16 - 9c^{10} $$\n\nFinal answer: The simplified expression is $$ \boxed{16 - 9c^{10}} $$