To solve the polynomial division [tex]\((4x^2 + 5x - 6) \div (x + 2)\)[/tex], we will use polynomial long division. Here is the step-by-step process:
1. Setup the Division:
Write the dividend [tex]\(4x^2 + 5x - 6\)[/tex] and the divisor [tex]\(x + 2\)[/tex] in the long division format.
```
__________
x + 2 | 4x^2 + 5x - 6
```
2. First Division:
Divide the first term of the dividend [tex]\(4x^2\)[/tex] by the first term of the divisor [tex]\(x\)[/tex]. This gives [tex]\(4x\)[/tex].
```
4x
__________
x + 2 | 4x^2 + 5x - 6
```
3. First Multiplication:
Multiply the entire divisor [tex]\(x + 2\)[/tex] by the result [tex]\(4x\)[/tex] and write it under the dividend.
```
4x
___________
x + 2 | 4x^2 + 5x - 6
-(4x^2 + 8x)
------------
-3x - 6
```
4. First Subtraction:
Subtract [tex]\(4x^2 + 8x\)[/tex] from [tex]\(4x^2 + 5x - 6\)[/tex]. This results in [tex]\(-3x - 6\)[/tex].
5. Second Division:
Divide the first term of the new polynomial [tex]\(-3x\)[/tex] by the first term of the divisor [tex]\(x\)[/tex]. This gives [tex]\(-3\)[/tex].
```
4x - 3
___________
x + 2 | 4x^2 + 5x - 6
-(4x^2 + 8x)
------------
-3x - 6
+(-3x - 6)
------------
0
```
6. Second Multiplication:
Multiply the entire divisor [tex]\(x + 2\)[/tex] by the result [tex]\(-3\)[/tex] and write it under [tex]\(-3x - 6\)[/tex].
```
4x - 3
___________
x + 2 | 4x^2 + 5x - 6
-(4x^2 + 8x)
------------
-3x - 6
-(-3x - 6)
------------
0
```
7. Second Subtraction:
Subtract [tex]\(-3x - 6\)[/tex] from [tex]\(-3x - 6\)[/tex]. This results in a remainder of [tex]\(0\)[/tex].
Thus, the quotient of the division is [tex]\(4x - 3\)[/tex] with a remainder of [tex]\(0\)[/tex].
So, the answer is:
A. [tex]\(4x - 3\)[/tex]
