To find the completely factored form of the expression d4 + 8d2 + 16, we will start by treating it as a quadratic in terms of d2.
We can rewrite the expression as follows:
(d2)2 + 8(d2) + 16
Next, we can let x = d2 so that the expression becomes:
x2 + 8x + 16
Now, we will factor this quadratic. We need to find two numbers that multiply to 16 (the constant term) and add up to 8 (the coefficient of the linear term). These two numbers are 4 and 4.
So, we can rewrite the quadratic as:
(x + 4)(x + 4) or (x + 4)2
Substituting back x = d2, we get:
(d2 + 4)2
Now, we know that the term d2 + 4 cannot be factored further using real numbers since it does not have real roots (as the discriminant is negative). Therefore, the completely factored form of d4 + 8d2 + 16 is:
(d2 + 4)2
Thus, the final answer is:
(d2 + 4)2