Divide :
`a^(2) + 7a + 12` by a + 4

Answer

To divide the polynomial \( a^2 + 7a + 12 \) by \( a + 4 \), we will use polynomial long division. Here are the steps:

### Step 1: Set up the division
We write \( a^2 + 7a + 12 \) as the dividend and \( a + 4 \) as the divisor.

### Step 2: Divide the leading terms
Divide the leading term of the dividend \( a^2 \) by the leading term of the divisor \( a \):
\[ \frac{a^2}{a} = a \]
This means the first term of our quotient is \( a \).

### Step 3: Multiply and subtract
Now, multiply \( a \) by the entire divisor \( a + 4 \):
\[ a(a + 4) = a^2 + 4a \]
Next, subtract this result from the original polynomial:
\[ (a^2 + 7a + 12) - (a^2 + 4a) = 7a - 4a + 12 = 3a + 12 \]

### Step 4: Repeat the process
Now, we repeat the process with the new polynomial \( 3a + 12 \). Divide the leading term \( 3a \) by the leading term of the divisor \( a \):
\[ \frac{3a}{a} = 3 \]
So, the next term of our quotient is \( 3 \).

### Step 5: Multiply and subtract again
Multiply \( 3 \) by the divisor \( a + 4 \):
\[ 3(a + 4) = 3a + 12 \]
Now, subtract this from \( 3a + 12 \):
\[ (3a + 12) - (3a + 12) = 0 \]

### Conclusion
Since the remainder is \( 0 \), we have completed the division. The final answer is:
\[ \frac{a^2 + 7a + 12}{a + 4} = a + 3 \]