From the question and options provided in the image, Bayani is trying to prove that the sum of two rational numbers is a rational number. The given steps and options suggest various approaches to continue the proof.

Here’s the breakdown:

1. Definition of Rational Numbers:
   $\frac{a}{b}$ and $\frac{c}{d}$ are rational numbers, meaning $a, b, c, d$ are integers, and $b \neq 0$, $d \neq 0$.

2. Adding Two Rational Numbers:
   The sum is given by:
   $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}.$

3. Best Method to Continue:
   The correct approach is to confirm that:

   • $ad + bc$ is an integer (sum of products of integers).
   • $bd$ is an integer (product of integers).
   • $\frac{ad + bc}{bd}$ is a rational number since the numerator and denominator are integers, and $bd \neq 0$.

Thus, the best choice is:
"Show that $ad, bc,$ and $bd$ are all integers."

Would you like a step-by-step verification of this reasoning or have any questions?

Here are 5 related questions to expand your understanding:

1. How is the closure property of integers under addition and multiplication used in this proof?
2. Why is it important to verify that $bd \neq 0$?
3. Can this proof be extended to subtraction of rational numbers?
4. How does this proof generalize to other operations like multiplication or division?
5. What changes if irrational numbers are involved in a similar proof?

Tip: Always confirm that the numerator and denominator in a rational number operation are integers, and the denominator is nonzero, as it ensures the result is rational.