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The problem given in the image can be formulated as a system of equations:

### Problem Description:
- John and Jivanti together have 45 marbles.
- Both of them lose 5 marbles each.
- The product of the number of marbles they now have is 124.
- We are tasked with finding how many marbles they initially had.

### Step-by-Step Solution:

1. **Let John have \( x \) marbles** initially, and let Jivanti have \( 45 - x \) marbles** (since together they have 45 marbles).

2. After losing 5 marbles each:
   - John has \( x - 5 \) marbles.
   - Jivanti has \( 45 - x - 5 = 40 - x \) marbles.

3. According to the problem, the product of the marbles they now have is 124:
   \[
   (x - 5)(40 - x) = 124
   \]

4. **Expand the equation**:
   \[
   x(40 - x) - 5(40 - x) = 124
   \]
   \[
   40x - x^2 - 200 + 5x = 124
   \]
   \[
   -x^2 + 45x - 200 = 124
   \]

5. **Rearrange the equation** to form a standard quadratic equation:
   \[
   -x^2 + 45x - 324 = 0
   \]
   Multiply by -1 to simplify:
   \[
   x^2 - 45x + 324 = 0
   \]

6. **Solve the quadratic equation** using the quadratic formula:
   \[
   x = \frac{-(-45) \pm \sqrt{(-45)^2 - 4(1)(324)}}{2(1)}
   \]
   \[
   x = \frac{45 \pm \sqrt{2025 - 1296}}{2}
   \]
   \[
   x = \frac{45 \pm \sqrt{729}}{2}
   \]
   \[
   x = \frac{45 \pm 27}{2}
   \]

7. **Find the solutions**:
   \[
   x = \frac{45 + 27}{2} = \frac{72}{2} = 36
   \]
   \[
   x = \frac{45 - 27}{2} = \frac{18}{2} = 9
   \]

### Conclusion:
John either initially had **36 marbles**, and Jivanti had \( 45 - 36 = 9 \) marbles, **or** John had **9 marbles**, and Jivanti had \( 45 - 9 = 36 \) marbles.

Would you like further details on any part of the solution, or do you have any other questions?

Here are 5 follow-up questions for deeper understanding:
1. How would the solution change if both had lost a different number of marbles?
2. Can we use a different method to solve this quadratic equation, such as factoring?
3. What does the discriminant of the quadratic formula tell us about the nature of the solutions?
4. How can we verify that both solutions satisfy the original problem?
5. What if the product of the remaining marbles was different from 124?

**Tip**: Always check your final answers by plugging them back into the original conditions to ensure they satisfy all parts of the problem.

John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.

This detailed step-by-step solution explains how to solve a quadratic equation where John and Jivanti start with 45 marbles, lose 5 each, and the product of their remaining marbles is 124. Suitable for students in Grades 8-10.

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