To simplify the expression $\frac{3x^4 + ax^3 + bx^2 + 4x + 5}{x^2 + 2}$, we need to perform polynomial division.

Step 1: Set up the division

We will divide the polynomial $3x^4 + ax^3 + bx^2 + 4x + 5$ by $x^2 + 2$.

Step 2: Divide the first term

Divide the leading term of the numerator by the leading term of the denominator: $\frac{3x^4}{x^2} = 3x^2$ This gives us the first term of the quotient, $3x^2$.

Step 3: Multiply and subtract

Now, multiply $3x^2$ by the entire denominator $x^2 + 2$: $3x^2 \times (x^2 + 2) = 3x^4 + 6x^2$ Subtract this from the original polynomial: $(3x^4 + ax^3 + bx^2 + 4x + 5) - (3x^4 + 6x^2) = ax^3 + (b - 6)x^2 + 4x + 5$

Step 4: Repeat the process

Now, repeat the process with the new polynomial $ax^3 + (b - 6)x^2 + 4x + 5$.

1. Divide the leading term: $\frac{ax^3}{x^2} = ax$ So, the next term in the quotient is $ax$.

2. Multiply and subtract: $ax \times (x^2 + 2) = ax^3 + 2ax$ Subtract: $(ax^3 + (b - 6)x^2 + 4x + 5) - (ax^3 + 2ax) = (b - 6)x^2 + (4 - 2a)x + 5$

Step 5: Continue the process

Repeat the steps until you have fully divided the polynomial or obtained a remainder that cannot be divided by the denominator.

The quotient so far is $3x^2 + ax + (b - 6)$, and the remainder is $(4 - 2a)x + 5$.

So, the division yields: $\frac{3x^4 + ax^3 + bx^2 + 4x + 5}{x^2 + 2} = 3x^2 + ax + (b - 6) + \frac{(4 - 2a)x + 5}{x^2 + 2}$

Final Answer:

The simplified expression is: $3x^2 + ax + (b - 6) + \frac{(4 - 2a)x + 5}{x^2 + 2}$

Would you like to go over the details of this process or have any questions?

Here are five related questions you might find useful:

1. How would the solution change if the denominator were $x^2 + x + 2$?
2. What happens if $a$ and $b$ are specific numbers, like $a = 2$ and $b = 3$?
3. Can you explain how polynomial long division compares to synthetic division?
4. How would you find the remainder if you only needed that?
5. Can you confirm the final remainder by plugging in a value for $x$?

Tip: Always double-check your multiplication and subtraction steps when doing polynomial division to avoid simple mistakes.