Factor completely. Hint: Factor out the GCF first.
$12x^2+48x+48$

• A. $12\(\left\)(x+2\(\right\))^2$
• B. $12\(\left\)(x+4\(\right\))^2$
• C. $12x\(\left\)(x+2\(\right\))^2$
• D. $12x\(\left\)(x+4\(\right\))^2$

Answer: (A)

Verified step by step guidance

1. Identify the general form of a perfect square trinomial, which is either \(a^{2} + 2ab + b^{2}\) or \(a^{2} - 2ab + b^{2}\), where the trinomial can be factored into \((a + b)^{2}\) or \((a - b)^{2}\) respectively.
2. Look at the given trinomial and check if the first and last terms are perfect squares. For example, verify if the first term is a perfect square like \(a^{2}\) and the last term is a perfect square like \(b^{2}\).
3. Check the middle term to see if it matches \$2ab\( or \)-2ab\(, where \)a\( and \)b$ are the square roots of the first and last terms respectively. This confirms the trinomial is a perfect square.
4. Once confirmed, write the factorization as \((a + b)^{2}\) if the middle term is positive, or \((a - b)^{2}\) if the middle term is negative, using the values of \(a\) and \(b\) found from the square roots.
5. Double-check your factorization by expanding \((a \\pm b)^{2}\) to ensure it matches the original trinomial exactly.