To factor the quadratic expression \( x^2 - 6x - 7 \), we look for two numbers that multiply to \(-7\) (the constant term) and add to \(-6\) (the coefficient of the linear term).
The two numbers that satisfy these conditions are:
- \( -7 \) and \( 1 \)
Now, we can write the expression as: \[ x^2 - 7x + 1x - 7 \]
Next, we can group the terms: \[ (x^2 - 7x) + (1x - 7) \]
Now, factor by grouping: \[ x(x - 7) + 1(x - 7) \]
Combining gives us: \[ (x + 1)(x - 7) \]
So, the factored form of \( x^2 - 6x - 7 \) is: \[ (x - 7)(x + 1) \]
You can check this factorization by expanding: \[ (x - 7)(x + 1) = x^2 + x - 7x - 7 = x^2 - 6x - 7 \]
Everything checks out! The correct factorization is: \[ (x - 7)(x + 1) \]