Question

factor $x^2 + 12x + 36$. the factored form is \boxed{}

Explanation:

Step1: Recall the perfect square trinomial formula

The perfect square trinomial formula is $a^2 + 2ab + b^2=(a + b)^2$.

Step2: Identify \(a\) and \(b\) in the given expression

For the expression \(x^2+12x + 36\), we have \(a = x\) (since \(x^2=a^2\)), and we need to find \(b\) such that \(2ab = 12x\) and \(b^2=36\).
Substituting \(a = x\) into \(2ab=12x\), we get \(2\times x\times b = 12x\). Dividing both sides by \(2x\) (assuming \(x
eq0\)), we find \(b = 6\).
We can also check \(b^2\): if \(b = 6\), then \(b^2=6^2 = 36\), which matches the constant term in the given trinomial.

Step3: Apply the perfect square trinomial formula

Using the formula \(a^2+2ab + b^2=(a + b)^2\) with \(a = x\) and \(b = 6\), we get \(x^2+12x + 36=(x + 6)^2\).

Answer:

$(x + 6)^2$