How do you factor completely \[77{x^2} + 102x + 16\]

Hint: By using the factorization method to obtain the factors of the given equation\[{x^2} + yx + z\]. Here we can use the mid term splitting method to factorise this equation in which we are going to split the middle term and can have required two factors.
For equation \[{x^2} + yx + z\]
Here we have to split “y” in such a way that the split number say “a” and “b” have the relationship \[a \times b = z(1)\] and \[a + b = y\] for the above equation.
 Now after splitting the number take the common factors in a bracket and rest in another bracket and will get the factors.

Formulae Used: We use a mid term splitting rule here as explained above in hint for getting the required factors needed as per the given equation in question above.

Complete step by step solution:
Here, for the given equation split “102” into two number following the condition of midterm splitting rule-
\[
  77{x^2} + 102x + 16 \\
  77{x^2} + (88 + 14)x + 16 \\
  77{x^2} + 88x + 14x + 16 \\
  11x(7x + 8) + 2(7x + 8) \\
  (7x + 8)(11x + 2) \\
 \]
So the factors for the above equation is \[(7x + 8),(11x +2)\]

Note: Mid term split method is easy to find factor, but another method that is taking the highest common factor directly and obtaining the factors. This method works sometimes in some questions only mostly for 3 variable questions.
In two variable questions it will work but make the question very complicated, so you have to be careful while using the highest common factor method. This method is fast and easy to use but is applicable to certain specific questions only.
Example of this method could be a three variable algebraic equation like:
\[
  xyz + xy \\
    \\
 \]