To factor the quadratic expression \( 2x^2 - 15x + 7 \), we will look for two numbers that multiply to \( 2 \times 7 = 14 \) (the product of the leading coefficient and the constant term) and add up to \(-15\) (the coefficient of \(x\)).
The two numbers that meet these criteria are \(-14\) and \(-1\):
- \(-14 \times -1 = 14\)
- \(-14 + -1 = -15\)
Next, we can rewrite the middle term of the quadratic expression using these numbers: \[ 2x^2 - 14x - 1x + 7 \]
Now we group the terms: \[ (2x^2 - 14x) + (-1x + 7) \]
Next, we factor out the greatest common factor from each group: \[ 2x(x - 7) - 1(x - 7) \]
Now we can factor by grouping: \[ (2x - 1)(x - 7) \]
Thus, the completely factored form of the expression \( 2x^2 - 15x + 7 \) is: \[ \boxed{(2x - 1)(x - 7)} \]