Write the quotient and remainder when we divide :
`(x^(2) + 12 x + 35)` by (x + 7)

Answer

To find the quotient and remainder when dividing the polynomial \( x^2 + 12x + 35 \) by \( x + 7 \), we can use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We want to divide \( x^2 + 12x + 35 \) by \( x + 7 \).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \( x^2 \) by the leading term of the divisor \( x \):
\[ \frac{x^2}{x} = x \]
This gives us the first term of the quotient.

### Step 3: Multiply and subtract
Now, multiply \( x \) by the entire divisor \( x + 7 \):
\[ x(x + 7) = x^2 + 7x \]
Subtract this from the original polynomial:
\[ (x^2 + 12x + 35) - (x^2 + 7x) = 12x - 7x + 35 = 5x + 35 \]

### Step 4: Repeat the process
Now, we need to divide \( 5x + 35 \) by \( x + 7 \). Divide the leading term \( 5x \) by \( x \):
\[ \frac{5x}{x} = 5 \]
This gives us the next term of the quotient.

### Step 5: Multiply and subtract again
Multiply \( 5 \) by the divisor \( x + 7 \):
\[ 5(x + 7) = 5x + 35 \]
Subtract this from \( 5x + 35 \):
\[ (5x + 35) - (5x + 35) = 0 \]

### Conclusion
Since the remainder is \( 0 \), we conclude that:
- **Quotient**: \( x + 5 \)
- **Remainder**: \( 0 \)

### Final Answer
The quotient is \( x + 5 \) and the remainder is \( 0 \).

---