To determine which of the given equations is a perfect square trinomial of the form [tex]\(x^2 + mx + m\)[/tex], we need to match it to the standard form of a perfect square trinomial, which is [tex]\((x + a)^2 = x^2 + 2ax + a^2\)[/tex].
### Step-by-Step Solution:
1. Compare each option to the form [tex]\((x + a)^2\)[/tex]:
- For [tex]\((x - 1)^2\)[/tex]:
[tex]\[
(x - 1)^2 = x^2 - 2x + 1
\][/tex]
Here, [tex]\(x^2 - 2x + 1\)[/tex]. It does not fit the form [tex]\(x^2 + mx + m\)[/tex].
- For [tex]\((x + 1)^2\)[/tex]:
[tex]\[
(x + 1)^2 = x^2 + 2x + 1
\][/tex]
Here, [tex]\(x^2 + 2x + 1\)[/tex]. It partially fits but does not carry the same coefficient for the middle term and last term.
- For [tex]\((x + 2)^2\)[/tex]:
[tex]\[
(x + 2)^2 = x^2 + 4x + 4
\][/tex]
Here, [tex]\(x^2 + 4x + 4\)[/tex]. It fits the form [tex]\(x^2 + mx + m\)[/tex] perfectly where [tex]\(m = 4\)[/tex].
- For [tex]\((x + 4)^2\)[/tex]:
[tex]\[
(x + 4)^2 = x^2 + 8x + 16
\][/tex]
Here, [tex]\(x^2 + 8x + 16\)[/tex]. It does not fit the form [tex]\(x^2 + mx + m\)[/tex].
2. Find the matching equation:
Among the given options, only [tex]\((x + 2)^2 = x^2 + 4x + 4\)[/tex] fits the needed form of [tex]\(x^2 + mx + m\)[/tex].
Thus, the correct equation must be [tex]\((x + 2)^2\)[/tex], which means:
[tex]\[
x^2 + mx + m = (x + 2)^2
\][/tex]
So, the correct option is:
[tex]\[
\boxed{3}
\][/tex]
