factorize the given expression `2x^2+11x-21`

Answer

To factorize the expression \(2x^2 + 11x - 21\), we will use the middle term splitting method. Here are the steps:

### Step-by-Step Solution:

1. **Identify the coefficients**:
- The coefficient of \(x^2\) is \(2\).
- The coefficient of \(x\) is \(11\).
- The constant term is \(-21\).

2. **Multiply the coefficient of \(x^2\) and the constant term**:
- Multiply \(2\) (coefficient of \(x^2\)) and \(-21\) (constant term):
\[ 2 \times -21 = -42 \]

3. **Find two numbers that multiply to \(-42\) and add to \(11\)**:
- We need two numbers that multiply to \(-42\) and add up to \(11\).
- The numbers are \(14\) and \(-3\) because:
\[ 14 \times -3 = -42 \quad \text{and} \quad 14 + (-3) = 11 \]

4. **Rewrite the middle term using these numbers**:
- Rewrite \(11x\) as \(14x - 3x\):
\[ 2x^2 + 14x - 3x - 21 \]

5. **Group the terms**:
- Group the first two terms and the last two terms:
\[ (2x^2 + 14x) + (-3x - 21) \]

6. **Factor out the common factors in each group**:
- From the first group \(2x^2 + 14x\), factor out \(2x\):
\[ 2x(x + 7) \]
- From the second group \(-3x - 21\), factor out \(-3\):
\[ -3(x + 7) \]

7. **Combine the factored terms**:
- Now we have:
\[ 2x(x + 7) - 3(x + 7) \]
- Factor out the common binomial \((x + 7)\):
\[ (x + 7)(2x - 3) \]

### Final Answer:
The factorized form of the expression \(2x^2 + 11x - 21\) is:
\[ (x + 7)(2x - 3) \]