1. Write the given equilibria and their constants

   For $A \rightleftharpoons B + C$ the equilibrium constant is $K_{eq}^{(1)} = \frac{[B][C]}{[A]}$

   For $B + C \rightleftharpoons P$ the equilibrium constant is $K_{eq}^{(2)} = \frac{[P]}{[B][C]}$

2. Add the two reactions

   Adding, $A \rightleftharpoons B + C$ $B + C \rightleftharpoons P$

   $A \rightleftharpoons P$

   The intermediate species $B$ and $C$ cancel out.

3. Use the rule for equilibrium constants

   When two equilibrium reactions are added, their equilibrium constants are multiplied.

   Therefore, for $A \rightleftharpoons P$ the equilibrium constant is $K = K_{eq}^{(1)} \cdot K_{eq}^{(2)}$

4. Direct verification by multiplying expressions

   $K_{eq}^{(1)}K_{eq}^{(2)} = \left(\frac{[B][C]}{[A]}\right)\left(\frac{[P]}{[B][C]}\right) = \frac{[P]}{[A]}$

   And this is exactly the equilibrium constant for $A \rightleftharpoons P$

5. Check options

   • A: $\dfrac{K_{eq}^{(1)}}{K_{eq}^{(2)}}$ ❌
   • B: $K_{eq}^{(1)} + K_{eq}^{(2)}$ ❌
   • C: $K_{eq}^{(2)} - K_{eq}^{(1)}$ ❌
   • D: $K_{eq}^{(1)}K_{eq}^{(2)}$ ✅

Hence, the correct option is D.