To factorize the quadratic expression [tex]\( x^2 + 2x - 8 \)[/tex], we first need to identify its roots or solutions. Here's a detailed, step-by-step method to solve the problem:
1. Identify the quadratic equation: The given polynomial is [tex]\( x^2 + 2x - 8 \)[/tex].
2. Recognize the standard form of a quadratic equation: A quadratic equation is typically written in the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. For the polynomial [tex]\( x^2 + 2x - 8 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 2 \)[/tex]
- [tex]\( c = -8 \)[/tex]
3. Use the quadratic formula to find the roots: The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Let's substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the quadratic formula.
4. Calculate the discriminant: The discriminant [tex]\( \Delta \)[/tex] is the part under the square root in the quadratic formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plugging in the coefficients:
[tex]\[
\Delta = 2^2 - 4(1)(-8) = 4 + 32 = 36
\][/tex]
5. Find the roots using the quadratic formula:
[tex]\[
x = \frac{-2 \pm \sqrt{36}}{2 \cdot 1}
\][/tex]
Simplifying inside the square root:
[tex]\[
\sqrt{36} = 6
\][/tex]
So, we have two possible roots:
[tex]\[
x_1 = \frac{-2 + 6}{2} = \frac{4}{2} = 2
\][/tex]
[tex]\[
x_2 = \frac{-2 - 6}{2} = \frac{-8}{2} = -4
\][/tex]
6. Write the factorization of the quadratic polynomial: Once we have the roots [tex]\( x_1 = 2 \)[/tex] and [tex]\( x_2 = -4 \)[/tex], the factorization of the polynomial [tex]\( x^2 + 2x - 8 \)[/tex] can be written as:
[tex]\[
(x - x_1)(x - x_2) = (x - 2)(x + 4)
\][/tex]
7. Match the factorization with the given options:
- A. [tex]\( (x+4)(x-2) \)[/tex]
- B. [tex]\( (x-8)(x+1) \)[/tex]
- C. [tex]\( (x+6)(x-4) \)[/tex]
- D. [tex]\( (x-4)(x+2) \)[/tex]
The correct factorization of [tex]\( x^2 + 2x - 8 \)[/tex] is [tex]\( (x + 4)(x - 2) \)[/tex], which corresponds to option A.
Therefore, the factorization of [tex]\( x^2 + 2x - 8 \)[/tex] is:
[tex]\[
\boxed{(x + 4)(x - 2)}
\][/tex]
