To use the polynomial identity \( x^2 + 2ax + a^2 = (x + a)(x + a) \) to rewrite \( x^2 + 4x + 4 \), we first identify the coefficients in the expression \( x^2 + 4x + 4 \).
Here, we have:
- The coefficient of \( x \) (which is \( 2a \)) is 4, so \( 2a = 4 \) ⇒ \( a = 2 \).
- The constant term \( a^2 \) is 4, so \( a^2 = 4 \) ⇒ \( a = 2 \).
So we can proceed with the polynomial identity by substituting \( a = 2 \) into the identity.
Now we rewrite \( x^2 + 4x + 4 \) using \( a = 2 \): \[ x^2 + 4x + 4 = (x + 2)(x + 2) = (x + 2)^2 \]
This establishes that: \[ x^2 + 4x + 4 = (x + 2)^2 \]
Thus, the polynomial \( x^2 + 4x + 4 \) can be expressed as \((x + 2)^2\).
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