1. **Problem statement:** We are given a parallelogram ABCD where the sum of the squares of two adjacent sides is $|AB|^2 + |AD|^2 = 12$. We need to find the sum of the squares of the diagonals, $|AC|^2 + |BD|^2$. 2. **Recall the parallelogram law:** For any parallelogram, the sum of the squares of the diagonals equals twice the sum of the squares of the adjacent sides: $$|AC|^2 + |BD|^2 = 2(|AB|^2 + |AD|^2)$$ 3. **Substitute the given value:** $$|AC|^2 + |BD|^2 = 2 \times 12 = 24$$ 4. **Answer:** The sum of the squares of the diagonals is $24$.