You need to divide the expression $(x^4 + 6x^2 - 2x + 6) \div (x^2 + 3)$.

Let’s go through the process step by step.

Step 1: Set up the long division

We need to divide the polynomial $x^4 + 6x^2 - 2x + 6$ by $x^2 + 3$. Polynomial long division is similar to numerical long division.

1. Divide the first term: $\frac{x^4}{x^2} = x^2$ So, the first term of the quotient is $x^2$.

2. Multiply and subtract: Multiply $x^2$ by the divisor $x^2 + 3$, which gives: $x^2(x^2 + 3) = x^4 + 3x^2$ Now subtract this from the original polynomial: $(x^4 + 6x^2 - 2x + 6) - (x^4 + 3x^2) = 3x^2 - 2x + 6$

Step 2: Continue dividing

Next, divide the first term of the new polynomial $3x^2 - 2x + 6$ by $x^2$: $\frac{3x^2}{x^2} = 3$ So, the next term in the quotient is $3$.

Step 3: Multiply and subtract again

Now multiply $3$ by the divisor $x^2 + 3$, which gives: $3(x^2 + 3) = 3x^2 + 9$ Subtract this from the current polynomial: $(3x^2 - 2x + 6) - (3x^2 + 9) = -2x - 3$

Step 4: Final result

Since $-2x - 3$ cannot be divided by $x^2 + 3$ (as its degree is lower), this is the remainder.

Final answer:

The result of the division is: $x^2 + 3 + \frac{-2x - 3}{x^2 + 3}$

This is the simplified result of the division.

Let me know if you need further clarifications!

Here are some related questions to explore:

1. How does polynomial long division compare to synthetic division?
2. What is the significance of the remainder in polynomial division?
3. Can you apply the same division method to different types of functions?
4. How would the quotient change if the divisor were different?
5. What happens when dividing polynomials with higher-degree terms?

Tip: When dividing polynomials, always ensure the dividend is written in descending powers of $x$.