To factor the quadratic expression [tex]\( x^2 - 4x - 5 \)[/tex], follow these steps:
1. Recognize that the quadratic expression is in the standard form [tex]\( ax^2 + bx + c \)[/tex], where in this case [tex]\( a = 1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = -5 \)[/tex].
2. We need to find two numbers that multiply to [tex]\( ac = -5 \)[/tex] and add to [tex]\( b = -4 \)[/tex].
3. These two numbers are [tex]\( -5 \)[/tex] and [tex]\( 1 \)[/tex], because:
- They multiply to [tex]\( -5 \times 1 = -5 \)[/tex].
- They add to [tex]\( -5 + 1 = -4 \)[/tex].
4. Rewrite the middle term [tex]\(-4x\)[/tex] using the two numbers [tex]\( -5 \)[/tex] and [tex]\( 1 \)[/tex]:
[tex]\[ x^2 - 5x + x - 5 \][/tex]
5. Factor by grouping:
[tex]\[
x^2 - 5x + x - 5 = x(x - 5) + 1(x - 5)
\][/tex]
6. Notice that [tex]\( (x - 5) \)[/tex] is a common factor:
[tex]\[
x(x - 5) + 1(x - 5) = (x - 5)(x + 1)
\][/tex]
Therefore, the factored form of the quadratic expression [tex]\( x^2 - 4x - 5 \)[/tex] is:
[tex]\[ (x - 5)(x + 1) \][/tex]
Thus, the correct option is:
[tex]\[ (x - 5)(x + 1) \][/tex]
