Determine the value we need to add to the equation to make it a perfect square trinomial, then factor it.
$x^2+8x+$__

• A. $16;{(x+4)}^2$
• B. $8;{(x+4)}^2$
• C. $16;\left(x+2\right)^2$
• D. $8;\left(x+2\right)^2$

Answer: (A)

Verified step by step guidance

1. Identify the given expression as \(x^2 + 8x + \_\), where we need to find the constant term to make it a perfect square trinomial.
2. Recall that a perfect square trinomial takes the form \(\left(x + a\right)^2 = x^2 + 2ax + a^2\). Here, the middle term \$8x\( corresponds to \)2ax\(, so set \)2a = 8\( to find \)a$.
3. Solve for \(a\) by dividing both sides by 2: \(a = \frac{8}{2} = 4\).
4. Calculate the constant term to add by squaring \(a\): \(a^2 = 4^2 = 16\). This is the value needed to complete the square.
5. Write the perfect square trinomial as \(x^2 + 8x + 16\) and factor it as \(\left(x + 4\right)^2\).