Question

Use the polynomial identity x2+2ax+a2=(x+a)(x+a) to rewrite x2+4x+4 . Which polynomial does this establish?(1 point)
Responses

x2+4x+4=(x−2)(x−2)
x squared plus 4 x plus 4 equals left parenthesis x minus 2 right parenthesis left parenthesis x minus 2 right parenthesis

x2+4x+4=(x−4)(x−4)
x squared plus 4 x plus 4 equals left parenthesis x minus 4 right parenthesis left parenthesis x minus 4 right parenthesis

x2+4x+4=(x+4)(x+4)
x squared plus 4 x plus 4 equals left parenthesis x plus 4 right parenthesis left parenthesis x plus 4 right parenthesis

x2+4x+4=(x+2)(x+2)

Answer 1

To rewrite \( x^2 + 4x + 4 \) using the polynomial identity \( x^2 + 2ax + a^2 = (x + a)(x + a) \), we will identify values for \( a \).

In the expression \( x^2 + 4x + 4 \):
- The coefficient of \( x \) is \( 4 \), which corresponds to \( 2a \). Therefore, \( 2a = 4 \) gives us \( a = 2 \).
- The last term \( a^2 \) should match \( 4 \). Since \( a = 2 \), we have \( a^2 = 2^2 = 4 \).

Thus, both conditions are satisfied. We can rewrite the polynomial as:

\[
x^2 + 4x + 4 = (x + 2)(x + 2)
\]

So the polynomial established is:

\[
x^2 + 4x + 4 = (x + 2)(x + 2)
\]

Therefore, the correct response is:

\( x^2 + 4x + 4 = (x + 2)(x + 2) \).