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The problem is asking to find the quotient and remainder using polynomial long division:

\[
\frac{x^6 + 5x^4 - 3x^2 - 15}{x^2 + 5}
\]

Let me walk you through the steps to solve it.

### Step 1: Divide the leading term of the dividend by the leading term of the divisor.
Divide \(x^6\) by \(x^2\), which gives \(x^4\).

### Step 2: Multiply \(x^4\) by the divisor \((x^2 + 5)\).
This gives:
\[
x^4(x^2 + 5) = x^6 + 5x^4
\]

### Step 3: Subtract this result from the original polynomial.
\[
(x^6 + 5x^4) - (x^6 + 5x^4) = 0
\]
The remaining polynomial is:
\[
-3x^2 - 15
\]

### Step 4: Divide the leading term of the new polynomial by the leading term of the divisor.
Divide \(-3x^2\) by \(x^2\), which gives \(-3\).

### Step 5: Multiply \(-3\) by the divisor.
\[
-3(x^2 + 5) = -3x^2 - 15
\]

### Step 6: Subtract this result from the remaining polynomial.
\[
(-3x^2 - 15) - (-3x^2 - 15) = 0
\]

### Final result:
The quotient is:
\[
x^4 - 3
\]
and the remainder is:
\[
0
\]

Thus:
- **Quotient**: \( x^4 - 3 \)
- **Remainder**: \( 0 \)

Would you like further clarification or details?

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Here are 5 related questions for you:
1. What happens if the remainder is not zero in polynomial division?
2. How can synthetic division be used in dividing polynomials?
3. How is the degree of the quotient related to the degrees of the dividend and divisor?
4. What are the applications of polynomial division in calculus?
5. How can you verify the result of a polynomial division?

**Tip:** Always check your polynomial division by multiplying the quotient by the divisor and adding the remainder, ensuring it equals the original dividend!

Find the quotient and remainder using long division for the expression (x^6 + 5x^4 - 3x^2 - 15) divided by (x^2 + 5).

Learn how to perform polynomial long division for the expression x^6 + 5x^4 - 3x^2 - 15 divided by x^2 + 5. This step-by-step solution explains how to find the quotient x^4 - 3 and remainder 0. Suitable for high school students.

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