To simplify the given expression, \(\frac{3x^4 + ax^3 + bx^2 + 4x + 5}{x^2 + 2}\), we can perform polynomial long division. This process will help us express the rational function in the form:

\[
\frac{3x^4 + ax^3 + bx^2 + 4x + 5}{x^2 + 2} = Q(x) + \frac{R(x)}{x^2 + 2}
\]

where \( Q(x) \) is the quotient and \( R(x) \) is the remainder.

Let's go through the division step-by-step:

1. **Divide the leading term of the numerator by the leading term of the denominator:**

   \[
   \frac{3x^4}{x^2} = 3x^2
   \]

   So, the first term in our quotient \( Q(x) \) is \( 3x^2 \).

2. **Multiply \( 3x^2 \) by the entire denominator \( x^2 + 2 \) and subtract from the numerator:**

   \[
   (3x^4 + ax^3 + bx^2 + 4x + 5) - (3x^4 + 6x^2) = (ax^3 + (b - 6)x^2 + 4x + 5)
   \]

   The \( 3x^4 \) terms cancel out.

3. **Repeat the process using the new polynomial \((ax^3 + (b - 6)x^2 + 4x + 5)\):**

   Now, we divide the leading term \( ax^3 \) by \( x^2 \):

   \[
   \frac{ax^3}{x^2} = ax
   \]

   The next term in our quotient is \( ax \).

4. **Multiply \( ax \) by \( x^2 + 2 \) and subtract from \((ax^3 + (b - 6)x^2 + 4x + 5)\):**

   \[
   (ax^3 + (b-6)x^2 + 4x + 5) - (ax^3 + 2ax) = ((b-6)x^2 + (4 - 2a)x + 5)
   \]

   The \( ax^3 \) terms cancel out.

5. **Continue with the new polynomial \((b-6)x^2 + (4-2a)x + 5\):**

   Divide the leading term \( (b-6)x^2 \) by \( x^2 \):

   \[
   \frac{(b-6)x^2}{x^2} = b-6
   \]

   The next term in our quotient is \( b-6 \).

6. **Multiply \( b-6 \) by \( x^2 + 2 \) and subtract from \( (b-6)x^2 + (4-2a)x + 5 \):**

   \[
   ((b-6)x^2 + (4-2a)x + 5) - ((b-6)x^2 + 2(b-6)) = (4-2a)x + (5 - 2(b-6))
   \]

   The \( (b-6)x^2 \) terms cancel out.

The remainder \( R(x) \) is now \( (4-2a)x + (5 - 2(b-6)) \).

**Final Result:**

\[
\frac{3x^4 + ax^3 + bx^2 + 4x + 5}{x^2 + 2} = 3x^2 + ax + (b-6) + \frac{(4-2a)x + (5 - 2(b-6))}{x^2 + 2}
\]

Would you like more details or have any questions about this? Here are some related questions to explore:

1. How do we determine the coefficients \(a\) and \(b\) to make the remainder zero?
2. How does synthetic division differ from polynomial long division?
3. How can we apply partial fraction decomposition to this expression?
4. What are the general rules for performing polynomial long division?
5. How can we use this method to find the roots of the numerator polynomial?

**Tip:** Always align terms by their degree when performing polynomial long division to avoid mistakes in the subtraction step.

Learn how to simplify the rational function (3x^4 + ax^3 + bx^2 + 4x + 5)/(x^2 + 2) using polynomial long division. This detailed solution covers step-by-step polynomial division, quotient and remainder expressions, and determining coefficients for a simplified form. Suitable for advanced high school students and beyond.

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