To solve this problem, we'll use polynomial division, which is similar to long division that you would use with numbers, but instead we're using polynomials.
Step 1:
We will start by dividing the highest degree term of the numerator by the highest degree term of the denominator. In our case, the highest degree term of the numerator is 2x^2 and of the denominator is x. Dividing 2x^2 by x gives us 2x.
Step 2:
Next, we multiply the entire denominator by the result we just got (2x), and subtract that from the original numerator. If we multiply (x + 3) by 2x, we get 2x^2 + 6x.
Step 3:
To subtract this from the original numerator, we write the result from step 2 underneath the numerator aligned by powers of x:
```
2x² + 10x + 12
- (2x² + 6x)
_______________
4x + 12
```
So, we end up with the new term 4x + 12.
Step 4:
Repeat step 1 with this new term as the numerator.
The highest degree term is 4x. Dividing 4x by x, which is the highest degree term of the original denominator (x + 3), we get 4.
Step 5:
We do the same procedure as step 2, but this time we multiply (x + 3) by 4, we get 4x + 12.
Step 6:
We now subtract this from the new term (4x + 12) obtained in step 3. The subtraction results end up in zero, which means the remainder is 0.
Step 7:
The quotient is obtained by adding up the results from step 1 (2x) and from step 4 (4), which gives: 2x + 4.
Therefore, the quotient of (2x^2 + 10x + 12) ÷ (x + 3) is 2x + 4 and the remainder is 0.