To tackle the problem of finding the product of [tex]\((x + 2)(x - 3)\)[/tex] using algebraic methods, we can employ the distributive property (also known as the FOIL method for binomials).
Here's a step-by-step solution:
1. Identify each term in the binomials:
[tex]\[
(x + 2)(x - 3)
\][/tex]
2. Distribute each term of the first binomial to each term of the second binomial:
- First, multiply the first terms in each binomial:
[tex]\[
x \cdot x = x^2
\][/tex]
- Next, multiply the outer terms:
[tex]\[
x \cdot (-3) = -3x
\][/tex]
- Then, multiply the inner terms:
[tex]\[
2 \cdot x = 2x
\][/tex]
- Finally, multiply the last terms in each binomial:
[tex]\[
2 \cdot (-3) = -6
\][/tex]
3. Combine all the products:
[tex]\[
x^2 + (-3x) + 2x + (-6)
\][/tex]
4. Combine like terms:
The like terms here are [tex]\(-3x\)[/tex] and [tex]\(2x\)[/tex]:
[tex]\[
x^2 - 3x + 2x - 6 = x^2 - x - 6
\][/tex]
Hence, the product of [tex]\((x + 2)(x - 3)\)[/tex] in its simplest form is:
[tex]\[
x^2 - x - 6
\][/tex]
