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To solve this division problem, we need to divide the polynomial \( 3x^4 + 3x^3 - 5x^2 + 3 \) by \( x + 5 \) using polynomial long division.

### Steps of Polynomial Long Division:

1. **Divide the leading terms**: Take the leading term of the dividend \(3x^4\) and divide it by the leading term of the divisor \(x\):
   \[
   \frac{3x^4}{x} = 3x^3
   \]
   This gives the first term of the quotient, \(3x^3\).

2. **Multiply and subtract**: Multiply \(3x^3\) by the entire divisor \(x + 5\):
   \[
   3x^3 \cdot (x + 5) = 3x^4 + 15x^3
   \]
   Subtract this from the original polynomial:
   \[
   (3x^4 + 3x^3 - 5x^2 + 3) - (3x^4 + 15x^3) = -12x^3 - 5x^2 + 3
   \]

3. **Repeat the process**: Now, divide the new leading term \(-12x^3\) by \(x\):
   \[
   \frac{-12x^3}{x} = -12x^2
   \]
   Multiply \(-12x^2\) by \(x + 5\) and subtract:
   \[
   -12x^2 \cdot (x + 5) = -12x^3 - 60x^2
   \]
   \[
   (-12x^3 - 5x^2 + 3) - (-12x^3 - 60x^2) = 55x^2 + 3
   \]

4. **Continue dividing**: Divide \(55x^2\) by \(x\):
   \[
   \frac{55x^2}{x} = 55x
   \]
   Multiply \(55x\) by \(x + 5\) and subtract:
   \[
   55x \cdot (x + 5) = 55x^2 + 275x
   \]
   \[
   (55x^2 + 3) - (55x^2 + 275x) = -275x + 3
   \]

5. **Last step**: Divide \(-275x\) by \(x\):
   \[
   \frac{-275x}{x} = -275
   \]
   Multiply \(-275\) by \(x + 5\) and subtract:
   \[
   -275 \cdot (x + 5) = -275x - 1375
   \]
   \[
   (-275x + 3) - (-275x - 1375) = 1378
   \]

### Final Answer:

The quotient is:
\[
3x^3 - 12x^2 + 55x - 275
\]
and the remainder is:
\[
1378
\]

So,
\[
\frac{3x^4 + 3x^3 - 5x^2 + 3}{x + 5} = 3x^3 - 12x^2 + 55x - 275 + \frac{1378}{x + 5}
\]

Would you like further details on any of the steps or have any questions?

### Additional Practice Questions:
1. Divide \(4x^4 - 2x^3 + x - 7\) by \(x + 2\).
2. Find the quotient and remainder when \(5x^3 - 4x^2 + x + 10\) is divided by \(x - 3\).
3. Perform polynomial long division on \(2x^5 - 3x^4 + 7x^3 - x + 6\) by \(x + 1\).
4. Use synthetic division to divide \(x^4 - 6x^3 + 11x^2 - 6x\) by \(x - 1\).
5. Verify the division of \(6x^3 + 11x^2 - 15x + 4\) by \(x + 2\) by multiplying back.

### Tip:
For polynomial division, always align terms by their degree and ensure each term in the dividend and divisor is accounted for to avoid mistakes.

Learn how to divide the polynomial 3x^4 + 3x^3 - 5x^2 + 3 by x + 5 using polynomial long division. This solution includes a step-by-step breakdown, covering important algebraic concepts suitable for Grades 10-12.

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