Divide the polynomials using long division. $$ \frac{x^{2}+7 x+12}{x+2} $$

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To divide the polynomial $$ x^{2}+7 x+12 $$ by $$ x+2 $$ using long division, we will follow these steps:

1. **Set up the long division**: Write $$ x^{2}+7 x+12 $$ under the long division symbol and $$ x+2 $$ outside.

2. **Divide the leading term**: Divide the leading term of the dividend $$ x^{2} $$ by the leading term of the divisor $$ x $$ to get $$ x $$.

3. **Multiply and subtract**: Multiply $$ x $$ by $$ x+2 $$ and subtract the result from the original polynomial.

4. **Repeat**: Bring down the next term and repeat the process until all terms are processed.

Let's perform these steps.

1. Divide $$ x^{2} $$ by $$ x $$ to get $$ x $$.
2. Multiply $$ x $$ by $$ x+2 $$ to get $$ x^{2}+2x $$.
3. Subtract $$ (x^{2}+2x) $$ from $$ (x^{2}+7x+12) $$:
   $$
   (x^{2}+7x+12) - (x^{2}+2x) = 5x + 12
   $$
4. Now, divide $$ 5x $$ by $$ x $$ to get $$ 5 $$.
5. Multiply $$ 5 $$ by $$ x+2 $$ to get $$ 5x + 10 $$.
6. Subtract $$ (5x + 10) $$ from $$ (5x + 12) $$:
   $$
   (5x + 12) - (5x + 10) = 2
   $$

Now we have completed the division. The quotient is $$ x + 5 $$ and the remainder is $$ 2 $$.

Thus, the result of the division is:
$$
\frac{x^{2}+7 x+12}{x+2} = x + 5 + \frac{2}{x+2}
$$
The division of $$ x^{2}+7x+12 $$ by $$ x+2 $$ results in $$ x + 5 $$ with a remainder of 2.

Divide the polynomials using long division.

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