To factor the quadratic expression \( x^2 + 32x + 256 \), we can look for two numbers that multiply to the constant term (256) and add to the linear coefficient (32).
In this case, the two numbers are 16 and 16 because:
- \( 16 \times 16 = 256 \)
- \( 16 + 16 = 32 \)
Thus, we can write the expression as:
\[ x^2 + 32x + 256 = (x + 16)(x + 16) \]
This can also be expressed as:
\[ (x + 16)^2 \]
Therefore, the factored form of \( x^2 + 32x + 256 \) is:
\[ (x + 16)^2 \]