### Video Transcript

Simplify the algebraic expression
π₯ minus two times π₯ plus six squared.

We are asked to expand and simplify
the product of a binomial expression with the square of a binomial expression. To do this, we first note that
multiplication is associative. Thus, we can evaluate the product
of these binomials by first multiplying two of the binomials together. Letβs start by expanding the square
of a binomial.

We can square π₯ plus six using the
horizontal method. We begin by writing π₯ plus six
squared in its expanded form, π₯ plus six times π₯ plus six. We can then distribute π₯ plus six
over π₯ plus six as follows. So, we have π₯ times π₯ plus six
plus six times π₯ plus six.

In each term, we have a product of
a monomial and a binomial. We can expand each term by
distributing the monomial factor over the binomial to get π₯ times π₯ plus π₯ times
six plus six times π₯ plus six times six. Now, we simplify each product, then
combine the like terms to get the trinomial π₯ squared plus 12π₯ plus 36. Thus, π₯ minus two times π₯ plus
six squared is equal to π₯ minus two times π₯ squared plus 12π₯ plus 36.

We can now distribute the factor π₯
minus two over the trinomial. In each term, we have the product
of a monomial and a binomial. We can expand each term by
distributing the monomial factor over the binomial. We do this by recalling the product
rule for exponents. This rule tells us that π₯ to the
power of π times π₯ to the power of π is equal to π₯ to the power of π plus
π.

And remember, whenever weβre
talking about monomials, weβre talking about our powers being nonnegative integer
values. So, in particular, the product rule
for monomials only counts when π and π are nonnegative integers, although this
formula is true for other values of π and π.

We continue by evaluating the
product for each term separately. Distributing π₯ squared gives π₯
cubed minus two π₯ squared. Distributing 12π₯ gives 12π₯
squared minus 24π₯. And distributing 36 gives 36π₯
minus 72. Finally, we combine like terms as
follows, which simplifies to π₯ cubed plus 10π₯ squared plus 12π₯ minus 72.

In conclusion, we have shown that
the algebraic expression π₯ minus two times π₯ plus six squared simplifies to π₯
cubed plus 10π₯ squared plus 12π₯ minus 72.