factorize the given expression `3x^2-14x+8`

Answer

To factorize the expression \(3x^2 - 14x + 8\), we will use the middle term splitting method. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The expression is in the form \(ax^2 + bx + c\), where:
- \(a = 3\)
- \(b = -14\)
- \(c = 8\)

### Step 2: Multiply \(a\) and \(c\)
We first multiply the coefficient of \(x^2\) (which is \(a\)) by the constant term \(c\):
\[ a \cdot c = 3 \cdot 8 = 24 \]

### Step 3: Find factors of 24
Next, we need to find two numbers that multiply to \(24\) (the result from Step 2) and add up to \(-14\) (the coefficient \(b\)). The factors of \(24\) that satisfy this condition are \(-12\) and \(-2\) because:
\[ -12 + (-2) = -14 \quad \text{and} \quad -12 \cdot -2 = 24 \]

### Step 4: Rewrite the expression
Now we can rewrite the middle term \(-14x\) using the factors we found:
\[ 3x^2 - 12x - 2x + 8 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (3x^2 - 12x) + (-2x + 8) \]

### Step 6: Factor out the common terms
Now we factor out the common factors from each group:
1. From the first group \(3x^2 - 12x\), we can factor out \(3x\):
\[ 3x(x - 4) \]
2. From the second group \(-2x + 8\), we can factor out \(-2\):
\[ -2(x - 4) \]

### Step 7: Combine the factors
Now we can combine the two groups:
\[ 3x(x - 4) - 2(x - 4) \]
Since both terms contain the common factor \((x - 4)\), we can factor that out:
\[ (x - 4)(3x - 2) \]

### Final Answer
Thus, the factorization of the expression \(3x^2 - 14x + 8\) is:
\[ \boxed{(x - 4)(3x - 2)} \]