To divide 6x³ - 2x² + 5x - 7 by x + 2, we can use polynomial long division. The quotient is ax² + bx + c and the remainder is d.
The steps for polynomial long division are:
1. Write the dividend (6x³ - 2x² + 5x - 7) and the divisor (x + 2) in long division form.
```
6x² - 14x + 33
x + 2 | 6x³ - 2x² + 5x - 7
- (6x³ + 12x²)
--------
-14x² + 5x
-(-14x² - 28x)
-------------
33x - 7
-(33x + 66)
---------
-73
```
2. The quotient is the result of the division of the leading term of the dividend by the leading term of the divisor. Therefore, a = 6x².
3. The next term of the quotient is found by multiplying the divisor by the first term of the quotient and subtracting the result from the dividend. Therefore, we can multiply (x + 2) by 6x² to get 6x³ + 12x², and subtract it from the dividend to get -14x² + 5x. The next term of the quotient is b = -14x.
4. We repeat the previous step to find the constant term of the quotient. Therefore, we can multiply (x + 2) by -14x to get -14x² - 28x, and subtract it from the dividend to get 33x - 7. The constant term of the quotient is c = 33.
5. The remainder is the final value in the long division process, which is -73. Therefore, d = -73.
Therefore, the quotient is 6x² - 14x + 33 and the remainder is -73.