Let's factor the polynomial [tex]\(x^2 + 2x - 3x - 6\)[/tex] by the method of double grouping. Follow these steps:
1. Group terms with common factors:
Simply gather the terms in pairs, ensuring each group has a common factor:
[tex]\[
\left(x^2 + 2x\right) + (-3x - 6)
\][/tex]
2. Factor the GCF from each group:
For the first group [tex]\((x^2 + 2x)\)[/tex]:
- The greatest common factor (GCF) is [tex]\(x\)[/tex].
- Factor out [tex]\(x\)[/tex]:
[tex]\[
x(x + 2)
\][/tex]
For the second group [tex]\((-3x - 6)\)[/tex]:
- The greatest common factor (GCF) is [tex]\(-3\)[/tex].
- Factor out [tex]\(-3\)[/tex]:
[tex]\[
-3(x + 2)
\][/tex]
Now, combine the factored expressions:
[tex]\[
x(x + 2) - 3(x + 2)
\][/tex]
3. Write the polynomial as a product of binomials:
Notice that both terms, [tex]\(x(x + 2)\)[/tex] and [tex]\(-3(x + 2)\)[/tex], have a common binomial factor [tex]\((x + 2)\)[/tex]. Factor out this common binomial factor:
[tex]\[
(x - 3)(x + 2)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\(x^2 + 2x - 3x - 6\)[/tex] is:
[tex]\[
(x - 3)(x + 2)
\][/tex]
