If the polynomial `(x^(4)+2x^(3)+8x^(2)+12x+18)` is divided by another polynomial `(x^(2)+5)`, the remainder comes out to be (px+q). Find the values of p and q.

Explanation

To solve the problem of finding the values of \( p \) and \( q \) when the polynomial \( x^4 + 2x^3 + 8x^2 + 12x + 18 \) is divided by \( x^2 + 5 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Polynomials**:
- Dividend (the polynomial to be divided): \( P(x) = x^4 + 2x^3 + 8x^2 + 12x + 18 \)
- Divisor (the polynomial by which we divide): \( D(x) = x^2 + 5 \)

2. **Perform Polynomial Long Division**:
- Divide \( P(x) \) by \( D(x) \).

3. **First Division**:
- To eliminate \( x^4 \), multiply \( D(x) \) by \( x^2 \):
\[ x^2 \cdot (x^2 + 5) = x^4 + 5x^2 \]
- Subtract this from \( P(x) \):
\[ (x^4 + 2x^3 + 8x^2 + 12x + 18) - (x^4 + 5x^2) = 2x^3 + 3x^2 + 12x + 18 \]

4. **Second Division**:
- Now, to eliminate \( 2x^3 \), multiply \( D(x) \) by \( 2x \):
\[ 2x \cdot (x^2 + 5) = 2x^3 + 10x \]
- Subtract this from the current polynomial:
\[ (2x^3 + 3x^2 + 12x + 18) - (2x^3 + 10x) = 3x^2 + 2x + 18 \]

5. **Third Division**:
- Next, to eliminate \( 3x^2 \), multiply \( D(x) \) by \( 3 \):
\[ 3 \cdot (x^2 + 5) = 3x^2 + 15 \]
- Subtract this from the current polynomial:
\[ (3x^2 + 2x + 18) - (3x^2 + 15) = 2x + 3 \]

6. **Determine the Remainder**:
- The remainder after division is \( R(x) = 2x + 3 \).

7. **Identify \( p \) and \( q \)**:
- The remainder \( R(x) \) is in the form \( px + q \).
- By comparing \( R(x) = 2x + 3 \) with \( px + q \):
- \( p = 2 \)
- \( q = 3 \)

### Final Values:
- \( p = 2 \)
- \( q = 3 \)