To factorize the given polynomial [tex]\(x^2 + 4x - 45\)[/tex], we follow these steps to find a pair of numbers that multiply to the constant term (-45) and add to the coefficient of the linear term (4).
1. Identify the constants:
The polynomial is [tex]\(x^2 + 4x - 45\)[/tex].
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = -45\)[/tex].
2. Find the product of [tex]\(a \cdot c\)[/tex]:
The product [tex]\(ac\)[/tex] is [tex]\(1 \cdot (-45) = -45\)[/tex].
3. Identify two numbers that multiply to [tex]\(-45\)[/tex] and add to [tex]\(4\)[/tex]:
We look for two numbers, [tex]\(m\)[/tex] and [tex]\(n\)[/tex], such that:
[tex]\[
m \cdot n = -45 \quad \text{and} \quad m + n = 4.
\][/tex]
4. Determine the correct pair:
By inspecting pairs of factors of [tex]\(-45\)[/tex], we find that [tex]\(9\)[/tex] and [tex]\(-5\)[/tex] work because:
[tex]\[
9 \cdot (-5) = -45 \quad \text{and} \quad 9 + (-5) = 4.
\][/tex]
5. Rewrite the middle term using [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
Rewrite [tex]\(4x\)[/tex] using [tex]\(9\)[/tex] and [tex]\(-5\)[/tex]:
[tex]\[
x^2 + 4x - 45 = x^2 + 9x - 5x - 45.
\][/tex]
6. Factor by grouping:
Group the terms to simplify:
[tex]\[
(x^2 + 9x) + (-5x - 45).
\][/tex]
Factor out the common factors in each group:
[tex]\[
x(x + 9) - 5(x + 9).
\][/tex]
7. Factor out the common binomial:
Factor [tex]\((x + 9)\)[/tex] out from both terms:
[tex]\[
(x - 5)(x + 9).
\][/tex]
Thus, the complete factorization of [tex]\(x^2 + 4x - 45\)[/tex] is [tex]\((x + 9)(x - 5)\)[/tex].
The correct answer choice is:
C. [tex]\((x + 9)(x - 5)\)[/tex]
