First of all, let's explain what trinomal is, trinomal means that that there are 3 terms in the expression, in the following example expressions, I will emphasize each term by coloring them differently .
In the first example, we can see that "x" and "1" are different terms, because they are separated by a sum.In the second, "zx" is the multiplication of "z" by "x", multiplications always stay on the same term, so "zx" only counts as one term, then z(32+4x+z^2) is also only one term, of course there are sums on the inside of the parenthesis, but that doesn't count, parenthesis no matter what they have inside, ALWAYS count as 1 term, and it is also multiplied by z, but remember that multiplications stay on the same term, and plus 30, remember the only arithmetical operations that can separate factors are the substraction and the sum!
After that, I counted a full fraction as a term, but why? simple, stop looking the fraction like that, instead look it this way ( "/"stands for division): (z+zxy+3x)/(8-3y), then you say "oh! those are two parenthesis", that's correct, and they are dividing, as division is the INVERSE operation of multiplication, it also only counts as 1 term. Now the square root is quite the same, it is like having √(2xy-2z), remember that only sums and substractions separate factors, so as the radical isn't one of those, it is just one term. Needless to talk about the one.
Okay, that was it about Terms, now let's go back to factorizing.
Imagine having the following expression(Fig. A). By just watching it I am convinced that the result is (Fig B.), but why? to know that the expression is a Perfect Square Trinomal, I use the method on (Fig. C), which consists on getting the square root of each of the side terms (ordered by "x" quantity), multiply them and them multiply them again by two, if that result is exactly the center term, then, it is a perfect square trinomals, if not, it is not. And if you looked carefully, you could have seen that you can get the factorized expression by just getting the square root of both side terms and put them into a parenthesis where they add and are powered by two.
Special cases:
If in an expression, the middle number is being subtracted(Fig. A), the only thing you must change is the symbol inside the parenthesis(Fig. B), and if it is inverted(Fig. C), you just have to factorize by "-1" and solve the equation normally(Fig D and E).
Sometimes this can show up to you(Fig. A), to solve it, you just have to follow the same method, remember the square root of x^4 is x^2, don't get confused.
This expression can be even further factorized, but we will see how to do that later.
