remainder) $$ \left(5 x^{4}+2 x^{3}+7 x^{2}+6 x-24\right) \div\left(x^{2}+3\right) $$

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To find the **remainder** when dividing the polynomial $$ 5x^{4} + 2x^{3} + 7x^{2} + 6x - 24 $$ by $$ x^{2} + 3 $$, we can perform **polynomial long division**. Here's a step-by-step breakdown:

### Step 1: Setup the Division

We are dividing:
$$ 5x^{4} + 2x^{3} + 7x^{2} + 6x - 24 $$
by:
$$ x^{2} + 3 $$

### Step 2: Divide the Leading Terms

1. **Divide** the leading term of the dividend by the leading term of the divisor:
   $$ \frac{5x^{4}}{x^{2}} = 5x^{2} $$
   
2. **Multiply** the entire divisor by this result:
   $$ 5x^{2} \times (x^{2} + 3) = 5x^{4} + 15x^{2} $$
   
3. **Subtract** this from the original polynomial:
   $$
   \begin{align*}
   (5x^{4} + 2x^{3} + 7x^{2} + 6x - 24) - (5x^{4} + 15x^{2}) &= 2x^{3} - 8x^{2} + 6x - 24
   \end{align*}
   $$

### Step 3: Repeat the Process

1. **Divide** the new leading term by the leading term of the divisor:
   $$ \frac{2x^{3}}{x^{2}} = 2x $$
   
2. **Multiply** the divisor by this result:
   $$ 2x \times (x^{2} + 3) = 2x^{3} + 6x $$
   
3. **Subtract**:
   $$
   \begin{align*}
   (2x^{3} - 8x^{2} + 6x - 24) - (2x^{3} + 6x) &= -8x^{2} - 24
   \end{align*}
   $$

### Step 4: Final Division

1. **Divide** the leading term:
   $$ \frac{-8x^{2}}{x^{2}} = -8 $$
   
2. **Multiply** the divisor:
   $$ -8 \times (x^{2} + 3) = -8x^{2} - 24 $$
   
3. **Subtract**:
   $$
   \begin{align*}
   (-8x^{2} - 24) - (-8x^{2} - 24) &= 0
   \end{align*}
   $$

### Conclusion

The **remainder** after performing the division is **0**. This means that $$ x^{2} + 3 $$ is a **factor** of $$ 5x^{4} + 2x^{3} + 7x^{2} + 6x - 24 $$.

**Final Answer:** The remainder is 0.
The remainder is 0.

remainder) \( \left(5 x^{4}+2 x^{3}+7 x^{2}+6 x-24\right) \div\left(x^{2}+3\right) \)

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