### Video Transcript

If two π₯ minus π¦ times two π₯
minus five π¦ times three π₯ plus two π¦ is equal to ππ₯ cubed plus ππ₯ squared π¦
plus ππ₯π¦ squared plus ππ¦ cubed, what are the values of π, π, π, and π?

On the left-hand side of the
equation, we have the product of three binomials. And on the right-hand side of the
equation, we have a polynomial. We can find a polynomial expression
equivalent to the left-hand side of the equation by expanding the product. Letβs start off by finding the
product of the first two factors. To do this, we can take the product
of each pair of terms from the first parentheses with the second parentheses and add
the results.

So, we can distribute the first
term from the first binomial through both terms in the second binomial, resulting in
two π₯ times two π₯ plus two π₯ times negative five π¦. Then, we can distribute the second
term from the first binomial through both terms in the second binomial, resulting in
negative π¦ times two π₯ plus negative π¦ times negative five π¦. Simplifying the results gives four
π₯ squared minus 10π₯π¦ minus two π₯π¦ plus five π¦ squared. Then, we can combine like terms to
get four π₯ squared minus 12π₯π¦ plus five π¦ squared. Now we can substitute this
expression into our product of three binomials. Then, we can expand the product by
distributing the trinomial over each term in the binomial.

For the first term, we have 12π₯
cubed minus 36π₯ squared π¦ plus 15π₯π¦ squared. And for the second term, we have
eight π₯ squared π¦ minus 24π₯π¦ squared plus 10π¦ cubed. Combining the like terms and
simplifying gives 12π₯ cubed minus 28π₯ squared π¦ minus nine π₯π¦ squared plus 10π¦
cubed. We are told this is equal to ππ₯
cubed plus ππ₯ squared π¦ plus ππ₯π¦ squared plus ππ¦ cubed. For the polynomials to be equal,
their coefficients must be equal. Thus, π equals 12, π equals
negative 28, π equals negative nine, and π equals 10.