1. **Stating the problem:** Develop the equality $$(7x + x^4) \cdot (7x - x^4)$$. 2. **Formula used:** This is a product of conjugates, which follows the identity $$ (a+b)(a-b) = a^2 - b^2 $$. 3. **Applying the formula:** Here, $a = 7x$ and $b = x^4$. $$ (7x + x^4)(7x - x^4) = (7x)^2 - (x^4)^2 $$ 4. **Calculating powers:** $$ (7x)^2 = 49x^2 $$ $$ (x^4)^2 = x^{8} $$ 5. **Final expression:** $$ 49x^2 - x^{8} $$ **Answer:** $49x^2 - x^{8}$