factorize the given expression `6x^2-5x-21`

Answer

To factorize the expression \(6x^2 - 5x - 21\), we can follow these steps:

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

### Step 2: Multiply \(a\) and \(c\)
We need to multiply the coefficient of \(x^2\) (which is \(6\)) by the constant term (which is \(-21\)):
\[ 6 \times (-21) = -126 \]

### Step 3: Find two numbers that multiply to \(-126\) and add to \(-5\)
We need to find two numbers that multiply to \(-126\) and add to \(-5\). The factors of \(-126\) that satisfy this condition are \(9\) and \(-14\) because:
\[ 9 + (-14) = -5 \]
\[ 9 \times (-14) = -126 \]

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

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

### Step 6: Factor by grouping
Now we factor out the common factors from each group:
\[ 3x(2x + 3) - 7(2x + 3) \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \((2x + 3)\):
\[ (2x + 3)(3x - 7) \]

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

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