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. $16;{(x+2)}^2$
• D. $8;{(x+2)}^2$

Answer: (A)

Verified step by step guidance

1. Identify the given expression: \(x^2 + 8x + \_\), where we need to find the missing 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 coefficient of \(x\) is 8, which corresponds to \$2a$.
3. Solve for \(a\) by setting \$2a = 8\(, which gives \)a = 4$.
4. Calculate the constant term to add by squaring \(a\): \(a^2 = 4^2 = 16\). So, add 16 to complete the trinomial.
5. Write the perfect square trinomial and factor it as \(x^2 + 8x + 16 = \left(x + 4\right)^2\).