To find the correct factor pairs of the polynomial given in the problem, we can substitute the polynomial with its likely factors based on the given options. We want to ensure that the polynomial [tex]\( x^2 - x - 6 \)[/tex] is correctly represented.
Given polynomial:
[tex]\[ x^2 - x - 6 \][/tex]
Let's rewrite our options and multiply them to check which factors produce the polynomial:
1. Factors: (x - 1) and (x + 3)
[tex]\[
(x - 1)(x + 3) = x(x + 3) - 1(x + 3) = x^2 + 3x - x - 3 = x^2 + 2x - 3
\][/tex]
2. Factors: (x + 1) and (x - 3)
[tex]\[
(x + 1)(x - 3) = x(x - 3) + 1(x - 3) = x^2 - 3x + x - 3 = x^2 - 2x - 3
\][/tex]
3. Factors: (x - 2) and (x + 3)
[tex]\[
(x - 2)(x + 3) = x(x + 3) - 2(x + 3) = x^2 + 3x - 2x - 6 = x^2 + x - 6
\][/tex]
4. Factors: (x + 2) and (x - 3)
[tex]\[
(x + 2)(x - 3) = x(x - 3) + 2(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6
\][/tex]
Now, let's see which one of these expressions matches the given polynomial [tex]\( x^2 - x - 6 \)[/tex]:
- The first pair, [tex]\((x - 1)(x + 3)\)[/tex], results in [tex]\( x^2 + 2x - 3 \)[/tex]
- The second pair, [tex]\((x + 1)(x - 3)\)[/tex], results in [tex]\( x^2 - 2x - 3 \)[/tex]
- The third pair, [tex]\((x - 2)(x + 3)\)[/tex], results in [tex]\( x^2 + x - 6 \)[/tex]
- The fourth pair, [tex]\((x + 2)(x - 3)\)[/tex], results in [tex]\( x^2 - x - 6 \)[/tex]
Thus, the correct pair of factors for the polynomial [tex]\( x^2 - x - 6 \)[/tex] is:
[tex]\[ (x + 2) \text{ and } (x - 3) \][/tex]
