Apologies everyone, MarkFL was correct there was a type in my handwriting

But what I do not quite understand is how we went from this equation __= (x^2 − 2 ⋅ 3 ⋅ x + 3^2) + 4 __ to this __(x − 3) 2 + 4__

Could you please clarify, before conintuing with any other problem I would like to fully understand this one.

What we have to recognize here is the following:

\(\displaystyle (a-b)^2=a^2-2ab+b^2\)

Now, we should observe that:

\(\displaystyle x^2-2\cdot3\cdot x+3^2\)

fits that form, where \(a=x\) and \(b=3\). But, then you might ask, well, how did you know to break 13 up into the sum of 9 and 4? So, let's go back to the original expression:

\(\displaystyle x^2-6x+13\)

Now, let's look at the coefficient of the linear term which is \(-6\). If we divide by 2 and square, we get:

\(\displaystyle \left(\frac{-6}{2}\right)^2=9\)

This tells us we need a 9 as part of the square we're trying to build, and so we break 13 into 9+4, giving us:

\(\displaystyle x^2-6x+9+4\)

And, we are assured the first 3 terms can be written as the square of a binomial.

Does that make sense?