Let's solve the problem step-by-step to determine which equation is equivalent to the given equation: [tex]\(x^2 + 16x = 22\)[/tex].
First, we'll rewrite the given equation in the form of a perfect square.
### Step 1: Add and subtract the same value to form a perfect square
Consider the given equation:
[tex]\[ x^2 + 16x = 22 \][/tex]
To form a perfect square, we need to add and subtract [tex]\(64\)[/tex], because [tex]\(\left(\frac{16}{2}\right)^2 = 64\)[/tex].
### Step 2: Rewrite the equation
[tex]\[ x^2 + 16x + 64 - 64 = 22 \][/tex]
[tex]\[ (x + 8)^2 - 64 = 22 \][/tex]
### Step 3: Move the constant term to the right side
[tex]\[ (x + 8)^2 - 64 = 22 \][/tex]
[tex]\[ (x + 8)^2 = 22 + 64 \][/tex]
[tex]\[ (x + 8)^2 = 86 \][/tex]
Given this transformation, we can see that the equation [tex]\( (x + 8)^2 = 86 \)[/tex] is equivalent to the given equation [tex]\( x^2 + 16x = 22 \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{(x+8)^2=86}
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{C}
\][/tex]
