Problem: A village in the northeast in India is suffering from a flood. A merchant decided to help them with food items, etc. So he collected some money from different persons which is represented by x^4 + x^3 + 8x^2 + ax + b. If the number of persons who contributed for this service is x^2 + 1, find the value of a and b.

Solution:
To find the value of a and b, we need to use the given information and solve the equation.

Step 1: Identify the Number of Contributors
From the given information, we know that the number of persons who contributed is x^2 + 1.

Step 2: Express the Total Collected Money
The total collected money is represented by the expression x^4 + x^3 + 8x^2 + ax + b.

Step 3: Divide Total Money by Number of Contributors
To find the average contribution per person, we divide the total collected money by the number of contributors.

Step 4: Express Average Contribution per Person
The average contribution per person can be expressed as (x^4 + x^3 + 8x^2 + ax + b) / (x^2 + 1).

Step 5: Simplify the Expression
To simplify the expression, we need to perform polynomial long division.

Step 6: Perform Polynomial Long Division
Performing polynomial long division, we get:

```
x^2 + 8
___________________________
x^2 + 1 | x^4 + x^3 + 8x^2 + ax + b
- (x^4 + x^2)
_______________
x^3 + 7x^2
- (x^3 + x)
_______________
6x^2 + ax
- (6x^2 + 6)
_________________
ax + (b - 6)
```

Step 7: Equate the Remainder to Zero
Since the average contribution per person should be a whole number, the remainder (ax + b - 6) obtained from the polynomial long division should be zero.

Step 8: Set the Remainder to Zero and Solve for a and b
Setting the remainder to zero, we get:

ax + b - 6 = 0

Solving this equation, we can find the values of a and b.

Step 9: Conclusion
The values of a and b can be determined by solving the equation ax + b - 6 = 0, which represents the condition for the average contribution per person to be a whole number.

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