Let's determine which expression is equivalent to [tex]\((x+1)(x+1)\)[/tex] by simplifying it step-by-step.
First, let's expand [tex]\((x+1)(x+1)\)[/tex]:
1. Use the distributive property (also known as the FOIL method for binomials) to multiply the terms:
- Multiply the first terms: [tex]\(x \cdot x = x^2\)[/tex]
- Multiply the outer terms: [tex]\(x \cdot 1 = x\)[/tex]
- Multiply the inner terms: [tex]\(1 \cdot x = x\)[/tex]
- Multiply the last terms: [tex]\(1 \cdot 1 = 1\)[/tex]
2. Summarize all the products:
[tex]\[
(x+1)(x+1) = x^2 + x + x + 1
\][/tex]
Combine like terms:
[tex]\[
x^2 + x + x + 1 = x^2 + 2x + 1
\][/tex]
Now let's compare this result with the given options:
1. [tex]\(2(x+1)\)[/tex]
- Distribute the 2: [tex]\(2 \cdot x + 2 \cdot 1 = 2x + 2\)[/tex]
- This is not equal to [tex]\(x^2 + 2x + 1\)[/tex].
2. [tex]\((x+1)^2\)[/tex]
- By definition, [tex]\((x+1)^2\)[/tex] means [tex]\((x+1)(x+1)\)[/tex].
- This simplifies to [tex]\(x^2 + 2x + 1\)[/tex], which matches our result.
3. [tex]\(x^2 + 1\)[/tex]
- This expression lacks the middle term [tex]\(2x\)[/tex].
- It is not equal to [tex]\(x^2 + 2x + 1\)[/tex].
4. [tex]\(x + 1^2\)[/tex]
- Interpreting this, [tex]\(1^2\)[/tex] is just 1, so it's the same as [tex]\(x + 1\)[/tex].
- This does not match [tex]\(x^2 + 2x + 1\)[/tex].
From the analysis, the correct equivalent expression is:
[tex]\[
(x + 1)^2
\][/tex]
