Step-by-step explanation:
First, let's look at some examples of what a perfect square trinomial looks like. [tex]x^2 + 16x + 64[/tex]
This trinomial is made from:
[tex](x+8)^2[/tex]
So for your second question (x^2 + ___x + 36), we need to work backwards, starting with the last number of the trinomial, 36. Think of two identical numbers that would make 36 if they got multiplied together. Or: √36. Either way, we get 6. So we can put this as a squared binomial.
[tex](x+6)^2[/tex]
Then, we could solve the binomial to get our middle number. (Use FOIL: Multiply the First terms, then Outer terms, then Inner terms, and Last terms)
[tex](x+6)(x+6)[/tex]
[tex]x^2+6x+6x+36\\x^2+12x+36[/tex]
As you can see, our middle number is 12x, and that is what goes into the blank.
Answer: 12x
