To completely factor the expression x2 + 8x + 16, we first need to identify if it can be factored into the product of two binomials. The approach we will take involves recognizing it as a perfect square trinomial.
A perfect square trinomial takes the form (a + b)2 = a2 + 2ab + b2.
In our expression, we can see:
- a2 = x2 (where a = x)
- b2 = 16 (where b = 4, since 42 = 16)
- 2ab = 8x (which fits since 2 * x * 4 = 8x)
Since all these criteria match, we can factor the expression as follows:
(x + 4)2
Thus, the completely factored form of the expression x2 + 8x + 16 is (x + 4)2.
For further analysis, we can verify the factorization by expanding it back:
(x + 4)(x + 4) = x2 + 4x + 4x + 16 = x2 + 8x + 16.
Hence, the factorization is correct!