### Video Transcript

Factorize fully 81𝑚 squared plus
18𝑚 plus one.

In this question, we are given an
algebraic expression and asked to fully factor the algebraic expression. We should start by analyzing the
expression we need to factor. We can note that the expression has
three terms and that every term is the product of constants and variables, where the
variables are raised to nonnegative integer exponents. So, this is a trinomial.

When trying to factor a polynomial,
we should start by checking to see if all of the terms share a common factor, since
we can take out any factor shared by all of the terms. However, in this case, the constant
term is one. So, the greatest common divisor of
all of the terms will only be one. We are not done yet, since we need
to factor the expression fully.

The next step is to see if the
trinomial resembles any special trinomials that we know how to factor. If we do this, we can notice that
it is similar to a perfect square. That is, 𝑎 plus 𝑏 all squared is
equal to 𝑎 squared plus two 𝑎𝑏 plus 𝑏 squared. To see this, we can note that both
the first and last term of the trinomial are perfect squares, since nine 𝑚 all
squared is 81𝑚 squared and one squared is one.

Further, we can notice that 18𝑚
can be rewritten as two times nine 𝑚 times one. This then allows us to rewrite the
expression in the form of a perfect square, with 𝑎 equal to nine 𝑚 and 𝑏 equal to
one. This allows us to factor the
expression by substituting 𝑎 equals nine 𝑚 and 𝑏 equals one into the perfect
square formula to obtain nine 𝑚 plus one all squared. We cannot factor this expression
any further since nine 𝑚 and one have a greatest common divisor of one.

We can check that this answer is
correct by expanding our answer to check that we get the original trinomial. If we did this, we would get the
original trinomial, confirming that we can factor the expression to obtain nine 𝑚
plus one all squared.