To determine the factorization of the quadratic expression [tex]\(x^2 + 10x + 21\)[/tex], we want to express it in the form [tex]\((x + a)(x + b)\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants.
1. Starting from the standard quadratic form [tex]\(x^2 + 10x + 21\)[/tex], we look for two numbers that both:
a. Multiply together to give the constant term [tex]\(21\)[/tex].
b. Add up to give the coefficient of the linear term, which is [tex]\(10\)[/tex].
2. Let's examine the pairs of factors of [tex]\(21\)[/tex]:
- [tex]\(1\)[/tex] and [tex]\(21\)[/tex]. Sum: [tex]\(1 + 21 = 22\)[/tex]
- [tex]\(3\)[/tex] and [tex]\(7\)[/tex]. Sum: [tex]\(3 + 7 = 10\)[/tex]
- [tex]\(-1\)[/tex] and [tex]\(-21\)[/tex]. Sum: [tex]\(-1 - 21 = -22\)[/tex]
- [tex]\(-3\)[/tex] and [tex]\(-7\)[/tex]. Sum: [tex]\(-3 - 7 = -10\)[/tex]
3. We see that the pair [tex]\(3\)[/tex] and [tex]\(7\)[/tex] multiply to [tex]\(21\)[/tex] and add up to [tex]\(10\)[/tex].
4. Therefore, the quadratic expression [tex]\(x^2 + 10x + 21\)[/tex] can be factored as:
[tex]\[
(x + 3)(x + 7)
\][/tex]
Given the potential answers:
- [tex]\((x + 3)(x + 7)\)[/tex]
- [tex]\((x + 4)(x + 6)\)[/tex]
- [tex]\((x + 6)(x + 15)\)[/tex]
- [tex]\((x + 7)(x + 14)\)[/tex]
The correct factorization is:
[tex]\[
(x + 3)(x + 7)
\][/tex]
