Let's simplify the expression [tex]\((x+2)(x+6)\)[/tex] and write the resulting expression in standard form. Here is a detailed, step-by-step solution:
1. Expand the Factored Expression:
The expression given is [tex]\((x+2)(x+6)\)[/tex]. To expand this, we'll apply the distributive property (also known as the FOIL method for binomials). This involves multiplying each term in the first binomial by each term in the second binomial:
[tex]\[
(x+2)(x+6) = x(x+6) + 2(x+6)
\][/tex]
2. Distribute Each Term:
Now, distribute [tex]\(x\)[/tex] and [tex]\(2\)[/tex] through the terms in the parentheses:
- First, distribute [tex]\(x\)[/tex]:
[tex]\[
x(x+6) = x \cdot x + x \cdot 6 = x^2 + 6x
\][/tex]
- Next, distribute [tex]\(2\)[/tex]:
[tex]\[
2(x+6) = 2 \cdot x + 2 \cdot 6 = 2x + 12
\][/tex]
3. Combine Like Terms:
After distribution, we combine the results:
[tex]\[
x^2 + 6x + 2x + 12
\][/tex]
Combine the [tex]\(x\)[/tex]-terms:
[tex]\[
x^2 + (6x + 2x) + 12 = x^2 + 8x + 12
\][/tex]
4. Write the Result in Standard Form:
The standard form of a polynomial is to write it in descending order of the exponents. The resulting expression from the expansion and simplification is:
[tex]\[
x^2 + 8x + 12
\][/tex]
This is the simplified form of the given factored expression [tex]\((x+2)(x+6)\)[/tex].
