factorize the given expression `10x^2-9x-7`

Answer

To factorize the expression \(10x^2 - 9x - 7\), we will use the middle term splitting method. Here are the steps:

### Step-by-Step Solution:

1. **Identify the coefficients**:
- Coefficient of \(x^2\) (a) = 10
- Coefficient of \(x\) (b) = -9
- Constant term (c) = -7

2. **Multiply the coefficient of \(x^2\) with the constant term**:
- Calculate \(a \times c = 10 \times (-7) = -70\).

3. **Find two numbers that multiply to \(-70\) and add to \(-9\)**:
- The factors of \(-70\) that add up to \(-9\) are \(-14\) and \(5\) (since \(-14 + 5 = -9\)).

4. **Rewrite the middle term using these factors**:
- Rewrite \(-9x\) as \(-14x + 5x\).
- The expression now becomes:
\[ 10x^2 - 14x + 5x - 7 \]

5. **Group the terms**:
- Group the first two terms and the last two terms:
\[ (10x^2 - 14x) + (5x - 7) \]

6. **Factor out the common factors from each group**:
- From the first group \(10x^2 - 14x\), factor out \(2x\):
\[ 2x(5x - 7) \]
- From the second group \(5x - 7\), factor out \(1\):
\[ 1(5x - 7) \]

7. **Combine the factored terms**:
- Now, we can write the expression as:
\[ 2x(5x - 7) + 1(5x - 7) \]
- This can be factored further:
\[ (5x - 7)(2x + 1) \]

### Final Factored Form:
Thus, the final factored form of the expression \(10x^2 - 9x - 7\) is:
\[ (5x - 7)(2x + 1) \]