### Video Transcript

Factor 16๐ฅ to the fourth power ๐ฆ squared plus four ๐ฆ squared ๐ง to the fourth
power by completing the square.

To begin, we will consider whether the two terms in the given polynomial have a
highest common factor, or HCF, which may contain variables, constants, or products
of variables and constants.

We determine that four ๐ฆ squared is the highest common factor of the two terms. By dividing each term by the HCF, we find the remaining terms in the parentheses to
be four ๐ฅ to the fourth power plus ๐ง to the fourth power. We want to factor this expression by completing the square. So, we need to manipulate it to include a perfect square trinomial in the form ๐
squared plus or minus two ๐๐ plus ๐ squared, which can be factored as ๐ plus or
minus ๐ squared.

In these trinomials, ๐ and ๐ may be variables, constants, or products of variables
and constants. In this example, if we take ๐ squared to be four ๐ฅ to the fourth power and ๐
squared to be ๐ง to the fourth power, then our value of ๐ is the square root of ๐
squared, which is equal to two ๐ฅ squared. And our value of ๐ is the square root of ๐ squared, which is equal to ๐ง
squared. Then, our middle term is equal to two ๐๐, or in some cases negative two ๐๐. Two ๐๐ comes out to two times two ๐ฅ squared times ๐ง squared, which is four ๐ฅ
squared ๐ง squared.

In our next step, we will introduce the two ๐๐ term into the original
expression. For any term we introduce into the expression, we must add the same term with the
opposite sign; this way, we are effectively adding zero, which does not change the
polynomial. In this case, the zero gets added to the polynomial in the form of four ๐ฅ squared ๐ง
squared minus four ๐ฅ squared ๐ง squared. Our expression with these new terms is four ๐ฆ squared times four ๐ฅ to the fourth
power plus four ๐ฅ squared ๐ง squared plus ๐ง to the fourth power minus four ๐ฅ
squared ๐ง squared. We can now factor the first three terms in the parentheses as a perfect square
trinomial, giving us two ๐ฅ squared plus ๐ง squared squared.

Now we have a difference of squares, since the expression within the parentheses is
being squared and four ๐ฅ squared ๐ง squared is a perfect square, specifically the
square of two ๐ฅ๐ง, where ๐ is in the first parentheses and ๐ is in the second
parentheses. Following the formula for factoring a difference of squares, we get two ๐ฅ squared
plus ๐ง squared minus two ๐ฅ๐ง times two ๐ฅ squared plus ๐ง squared plus two
๐ฅ๐ง.

We then check whether the resulting polynomials within each set of parentheses can be
factored. In this case, both polynomials are prime. Therefore, we have that four ๐ฆ squared times two ๐ฅ squared minus two ๐ฅ๐ง plus ๐ง
squared times two ๐ฅ squared plus two ๐ฅ๐ง plus ๐ง squared represents the full
factorization of 16๐ฅ to the fourth power ๐ฆ squared plus four ๐ฆ squared ๐ง to the
fourth power.