Which of the following is the factored form of the expression `3x^2 + 5x - 12` ?

Options

A. `(3x-4)(x+3)`
B. `(3x+ 4)(x-3)`
C. `(3x-6)(x+2)`
D. `(3x+6)(x-2)`

Answer

To factor the expression \(3x^2 + 5x - 12\), we can follow these steps:

### Step 1: Rewrite the middle term
We need to express the middle term \(5x\) as the sum of two terms whose coefficients multiply to give the product of the coefficient of \(x^2\) (which is \(3\)) and the constant term (which is \(-12\)).

The product is:
\[ 3 \times -12 = -36 \]
We need two numbers that add up to \(5\) and multiply to \(-36\). The numbers \(9\) and \(-4\) satisfy this condition because:
\[ 9 + (-4) = 5 \quad \text{and} \quad 9 \times (-4) = -36 \]

So, we can rewrite \(5x\) as \(9x - 4x\):
\[ 3x^2 + 9x - 4x - 12 \]

### Step 2: Group the terms
Now, we can group the terms in pairs:
\[ (3x^2 + 9x) + (-4x - 12) \]

### Step 3: Factor out the common terms
Next, we factor out the common factors from each group:
1. From the first group \(3x^2 + 9x\), we can factor out \(3x\):
\[ 3x(x + 3) \]
2. From the second group \(-4x - 12\), we can factor out \(-4\):
\[ -4(x + 3) \]

Now, we have:
\[ 3x(x + 3) - 4(x + 3) \]

### Step 4: Factor out the common binomial
Notice that \((x + 3)\) is a common factor:
\[ (x + 3)(3x - 4) \]

### Final Factored Form
Thus, the factored form of the expression \(3x^2 + 5x - 12\) is:
\[ (x + 3)(3x - 4) \]

### Conclusion
The final answer is:
\[ (x + 3)(3x - 4) \]