1. **State the problem:** Simplify the expression $$\left(\frac{1}{2} a^2 b - 1\right)\left(\frac{1}{2} a^2 b + 1\right) + \frac{3}{4} a^4 b^2 - (a^4 - 1)(b^2 + 1).$$ 2. **Recall the formula:** The product of conjugates formula is $$ (x - y)(x + y) = x^2 - y^2. $$ Here, let $$x = \frac{1}{2} a^2 b$$ and $$y = 1.$$ 3. **Apply the formula:** $$\left(\frac{1}{2} a^2 b - 1\right)\left(\frac{1}{2} a^2 b + 1\right) = \left(\frac{1}{2} a^2 b\right)^2 - 1^2 = \frac{1}{4} a^4 b^2 - 1.$$ 4. **Rewrite the expression:** $$\frac{1}{4} a^4 b^2 - 1 + \frac{3}{4} a^4 b^2 - (a^4 - 1)(b^2 + 1).$$ 5. **Combine like terms:** $$\frac{1}{4} a^4 b^2 + \frac{3}{4} a^4 b^2 = \cancel{\frac{1}{4} a^4 b^2} + \cancel{\frac{3}{4} a^4 b^2} = a^4 b^2.$$ 6. **Expand the product:** $$(a^4 - 1)(b^2 + 1) = a^4 b^2 + a^4 - b^2 - 1.$$ 7. **Substitute back:** $$a^4 b^2 - 1 - (a^4 b^2 + a^4 - b^2 - 1) = a^4 b^2 - 1 - a^4 b^2 - a^4 + b^2 + 1.$$ 8. **Simplify by canceling terms:** $$\cancel{a^4 b^2} - 1 - \cancel{a^4 b^2} - a^4 + b^2 + 1 = -a^4 + b^2 + ( -1 + 1 ) = -a^4 + b^2.$$ **Final answer:** $$\boxed{-a^4 + b^2}.$$