To divide the polynomial [tex]\(4x^2 + 5x - 6\)[/tex] by [tex]\(x + 2\)[/tex], we can use polynomial long division. Let's go through the steps:
### Step 1: Set up the division
We want to divide:
[tex]\[ 4x^2 + 5x - 6 \][/tex]
by:
[tex]\[ x + 2 \][/tex]
### Step 2: Divide the leading terms
Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[ \frac{4x^2}{x} = 4x \][/tex]
### Step 3: Multiply and subtract
Multiply [tex]\(4x\)[/tex] by the divisor [tex]\(x + 2\)[/tex]:
[tex]\[ 4x \cdot (x + 2) = 4x^2 + 8x \][/tex]
Subtract this from the original polynomial:
[tex]\[ (4x^2 + 5x - 6) - (4x^2 + 8x) = (4x^2 + 5x - 6) - 4x^2 - 8x = -3x - 6 \][/tex]
### Step 4: Repeat the process
Repeat the process with [tex]\(-3x - 6\)[/tex]:
1. Divide the leading term [tex]\(-3x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{-3x}{x} = -3 \][/tex]
2. Multiply [tex]\(-3\)[/tex] by the divisor [tex]\(x + 2\)[/tex]:
[tex]\[ -3 \cdot (x + 2) = -3x - 6 \][/tex]
3. Subtract this from the current remainder:
[tex]\[ (-3x - 6) - (-3x - 6) = 0 \][/tex]
### Step 5: Conclusion
Since the remainder is 0, the division results in a quotient of [tex]\(4x - 3\)[/tex] and a remainder of 0.
Thus, the solution to the division of [tex]\(4x^2 + 5x - 6\)[/tex] by [tex]\(x + 2\)[/tex] is:
[tex]\[ \boxed{(4x - 3, 0)} \][/tex]
