Find the HCF of `15x^(2) + 8 x - 12, 3x^(2) + x - 2,3x^(2) - 2x , 9x^(2) - 12 x + 4`

Options

A. `3x -4`
B. `x-2`
C. `3x - 2`
D. `x - 4`

Answer

To find the HCF (Highest Common Factor) of the given polynomials \(15x^2 + 8x - 12\), \(3x^2 + x - 2\), \(3x^2 - 2x\), and \(9x^2 - 12x + 4\), we will factor each polynomial and then identify the common factors.

### Step 1: Factor the first polynomial \(15x^2 + 8x - 12\)

1. We need to express \(15x^2 + 8x - 12\) in a factored form.
2. Multiply the coefficient of \(x^2\) (which is 15) by the constant term (which is -12):
\[ 15 \times -12 = -180 \]
3. We need two numbers that multiply to \(-180\) and add up to \(8\). These numbers are \(18\) and \(-10\).
4. Rewrite the polynomial:
\[ 15x^2 + 18x - 10x - 12 \]
5. Group the terms:
\[ (15x^2 + 18x) + (-10x - 12) \]
6. Factor by grouping:
\[ 3x(5x + 6) - 2(5x + 6) \]
7. Factor out the common factor:
\[ (5x + 6)(3x - 2) \]

### Step 2: Factor the second polynomial \(3x^2 + x - 2\)

1. Multiply the coefficient of \(x^2\) (which is 3) by the constant term (which is -2):
\[ 3 \times -2 = -6 \]
2. We need two numbers that multiply to \(-6\) and add up to \(1\). These numbers are \(3\) and \(-2\).
3. Rewrite the polynomial:
\[ 3x^2 + 3x - 2x - 2 \]
4. Group the terms:
\[ (3x^2 + 3x) + (-2x - 2) \]
5. Factor by grouping:
\[ 3x(x + 1) - 2(x + 1) \]
6. Factor out the common factor:
\[ (x + 1)(3x - 2) \]

### Step 3: Factor the third polynomial \(3x^2 - 2x\)

1. Factor out the common term \(x\):
\[ x(3x - 2) \]

### Step 4: Factor the fourth polynomial \(9x^2 - 12x + 4\)

1. Multiply the coefficient of \(x^2\) (which is 9) by the constant term (which is 4):
\[ 9 \times 4 = 36 \]
2. We need two numbers that multiply to \(36\) and add up to \(-12\). These numbers are \(-6\) and \(-6\).
3. Rewrite the polynomial:
\[ 9x^2 - 6x - 6x + 4 \]
4. Group the terms:
\[ (9x^2 - 6x) + (-6x + 4) \]
5. Factor by grouping:
\[ 3x(3x - 2) - 2(3x - 2) \]
6. Factor out the common factor:
\[ (3x - 2)(3x - 2) \text{ or } (3x - 2)^2 \]

### Step 5: Identify the common factor

Now, we have the factors of each polynomial:
- \(15x^2 + 8x - 12 = (5x + 6)(3x - 2)\)
- \(3x^2 + x - 2 = (x + 1)(3x - 2)\)
- \(3x^2 - 2x = x(3x - 2)\)
- \(9x^2 - 12x + 4 = (3x - 2)^2\)

The common factor across all four polynomials is \(3x - 2\).

### Final Answer
The HCF of the given polynomials is:
\[ \boxed{3x - 2} \]