Use factor between to factorise `6x^(3) +17x^(2) +4x-12` completely.

Answer

To factorize the polynomial \(6x^3 + 17x^2 + 4x - 12\) completely, we will follow these steps:

### Step 1: Identify the polynomial
Let \(f(x) = 6x^3 + 17x^2 + 4x - 12\).

### Step 2: Use the Factor Theorem
We will check if \(x + 2\) is a factor by substituting \(x = -2\) into \(f(x)\).

### Step 3: Substitute \(x = -2\)
Calculate \(f(-2)\):
\[ f(-2) = 6(-2)^3 + 17(-2)^2 + 4(-2) - 12 \]
\[ = 6(-8) + 17(4) - 8 - 12 \]
\[ = -48 + 68 - 8 - 12 \]
\[ = -48 + 68 - 20 \]
\[ = 0 \]
Since \(f(-2) = 0\), \(x + 2\) is indeed a factor.

### Step 4: Perform Polynomial Long Division
Now, we will divide \(f(x)\) by \(x + 2\).

1. Divide the leading term: \(6x^3 \div x = 6x^2\).
2. Multiply \(6x^2\) by \(x + 2\):
\[ 6x^2(x + 2) = 6x^3 + 12x^2 \]
3. Subtract this from \(f(x)\):
\[ (6x^3 + 17x^2 + 4x - 12) - (6x^3 + 12x^2) = 5x^2 + 4x - 12 \]

4. Now, divide the leading term: \(5x^2 \div x = 5x\).
5. Multiply \(5x\) by \(x + 2\):
\[ 5x(x + 2) = 5x^2 + 10x \]
6. Subtract this from the previous result:
\[ (5x^2 + 4x - 12) - (5x^2 + 10x) = -6x - 12 \]

7. Divide the leading term: \(-6x \div x = -6\).
8. Multiply \(-6\) by \(x + 2\):
\[ -6(x + 2) = -6x - 12 \]
9. Subtract this:
\[ (-6x - 12) - (-6x - 12) = 0 \]

Thus, we have:
\[ f(x) = (x + 2)(6x^2 + 5x - 6) \]

### Step 5: Factor the quadratic \(6x^2 + 5x - 6\)
We will factor \(6x^2 + 5x - 6\) using the method of splitting the middle term.

1. Multiply the coefficient of \(x^2\) (which is 6) by the constant term (which is -6):
\[ 6 \times -6 = -36 \]
2. We need two numbers that multiply to -36 and add to 5. The numbers are 9 and -4.

3. Rewrite the middle term:
\[ 6x^2 + 9x - 4x - 6 \]

4. Group the terms:
\[ (6x^2 + 9x) + (-4x - 6) \]
5. Factor by grouping:
\[ 3x(2x + 3) - 2(2x + 3) \]
6. Factor out the common term:
\[ (3x - 2)(2x + 3) \]

### Step 6: Write the complete factorization
Now, we can write the complete factorization of the polynomial:
\[ f(x) = (x + 2)(3x - 2)(2x + 3) \]

### Final Answer
The complete factorization of \(6x^3 + 17x^2 + 4x - 12\) is:
\[ (x + 2)(3x - 2)(2x + 3) \]