`21x^2+5x-6`

Answer

To factor the polynomial \(21x^2 + 5x - 6\), we can follow these steps:

### Step 1: Identify the coefficients
The polynomial is in the form \(ax^2 + bx + c\), where:
- \(a = 21\)
- \(b = 5\)
- \(c = -6\)

### Step 2: Multiply \(a\) and \(c\)
We need to multiply \(a\) and \(c\):
\[ ac = 21 \times (-6) = -126 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(-126\) and add to \(5\). After checking pairs of factors, we find:
- \(14\) and \(-9\) work because:
\[ 14 \times (-9) = -126 \quad \text{and} \quad 14 + (-9) = 5 \]

### Step 4: Rewrite the middle term
Now we can rewrite the polynomial by breaking the middle term \(5x\) into \(14x - 9x\):
\[ 21x^2 + 14x - 9x - 6 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (21x^2 + 14x) + (-9x - 6) \]

### Step 6: Factor out the common factors from each group
From the first group \(21x^2 + 14x\), we can factor out \(7x\):
\[ 7x(3x + 2) \]
From the second group \(-9x - 6\), we can factor out \(-3\):
\[ -3(3x + 2) \]

### Step 7: Combine the factored terms
Now we can combine the two factored parts:
\[ 7x(3x + 2) - 3(3x + 2) \]
We can factor out the common binomial \((3x + 2)\):
\[ (3x + 2)(7x - 3) \]

### Final Answer
Thus, the factorization of the polynomial \(21x^2 + 5x - 6\) is:
\[ (3x + 2)(7x - 3) \]

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