To rewrite the polynomial [tex]\(x^2 + 4x + 4\)[/tex] using the given polynomial identity [tex]\(x^2 + 2ax + a^2 = (x + a)(x + a)\)[/tex], we need to identify the value of [tex]\(a\)[/tex] that makes the polynomial fit this form.
Given the polynomial [tex]\(x^2 + 4x + 4\)[/tex]:
1. We recognize that the polynomial [tex]\(x^2 + 4x + 4\)[/tex] must be rewritten in the form [tex]\(x^2 + 2ax + a^2\)[/tex].
2. Next, we compare the given polynomial [tex]\(x^2 + 4x + 4\)[/tex] with the form [tex]\(x^2 + 2ax + a^2\)[/tex] to find [tex]\(a\)[/tex]:
- The term [tex]\( 2a \)[/tex] in the identity should match the coefficient of the [tex]\(x\)[/tex]-term in the polynomial. Here, [tex]\(2ax = 4x\)[/tex].
- By comparing, we get [tex]\(2a = 4\)[/tex]. Solving for [tex]\(a\)[/tex] gives [tex]\(a = 2\)[/tex].
- Additionally, we check the constant term [tex]\(a^2\)[/tex] to confirm. Here, [tex]\(a^2 = 2^2 = 4\)[/tex], which matches the constant term in the polynomial.
Since the value of [tex]\(a\)[/tex] we found is 2, we can now rewrite the polynomial using the identity:
[tex]\[
x^2 + 4x + 4 = (x + 2)(x + 2)
\][/tex]
Therefore, the correct polynomial established is:
[tex]\[
x^2 + 4x + 4 = (x + 2)(x + 2)
\][/tex]
The answer is:
[tex]\[
x^2 + 4x + 4 = (x + 2)(x + 2)
\][/tex]
Thus, the correct option is:
[tex]\[ \boxed{x^2 + 4x + 4 = (x + 2)(x + 2)} \][/tex]
