We want to multiply and simplify each product.
First we have negative X times the quantity 2X plus five,
so we have a monomial times a binomial.
To find this product, we multiply the monomial by both terms
inside the parentheses,
which is called distribution.
We want to distribute the negative X,
which means you want to multiply negative X and positive 2X,
as well as negative X and positive five.
If it's helpful,
we can think of this as negative 1X to the first,
and the exponent X here is also a one.
So the first product is negative X times 2X,
or if we want negative 1X to the first
times 2X to the first
which would be negative two X to the second.
The next product is going to be negative X
times positive five,
which would be negative 5X, so we write minus 5X.
These are not like terms,
so this is our simplified product.
And we have negative 2X squared minus 5X.
Next, we have X times the quantity X squared
plus 7X minus four.
So we have a monomial times a trinomial.
We want to distribute the X,
so notice out here we'll have three products,
one, two, three.
The exponent on this X is one,
and the coefficient would also be one.
So our first product is X times X squared,
or if we want, X to the first times X squared,
that would be X to the third.
Next, we have X times 7X, or if we want,
X to the first times 7X to the first,
that would be positive 7X squared,
so we write plus 7X squared.
Our last product is X times negative four,
that would be negative 4X, so we write minus 4X.
Again, no like terms, so this is our simplified product.
Next we have a binomial times a binomial.
We have the quantity X plus eight
times the quantity X minus six.
Sometimes we'll refer to this multiplication as foiling.
I prefer to think of this as double distribution.
To find this product,
we need to distribute this first X into the second binomial,
so we'll have one, two products here,
and then we also need to distribute the second term
of the first binomial into the second binomial.
So we'll have one, two products here.
Notice how we have a total of four products.
And again, it's helpful
if we can think of this as X to the first
and this as X to the first.
And again, both coefficients would be one.
So let's keep track of our products,
X times X, or X to the first
times X to the first is X squared.
Next we have X times negative six, that's negative 6X.
We write minus 6X.
Our next product is eight times X, that's positive 8X.
We write plus 8X.
And our last product is eight times negative 6,
that's negative 48, so we write minus 48.
But notice here we do have two like terms,
we have two X terms here, so we need to combine these.
We have X squared, and then negative 6X plus 8X
is positive 2X, so plus 2X minus 48.
This is our simplified product.
Now for our last example, we have the quantity
X plus nine squared, so we're squaring a binomial.
There are no shortcuts here.
We cannot just square both terms.
We need to write out two factors of X plus nine
and multiply like we normally would.
So we have double distribution here,
so we'll have four products.
We distribute this X, one, two products there,
and then we distribute the nine,
so we have two products here,
a total of four products.
Again, this is X to the first, and this is X to the first.
So our first product is X times X, that's X squared,
and then we have X times nine,
that's positive 9X, so plus 9X.
Next, we have nine times X, so that's plus 9X.
And then finally we have nine times nine, that's 81,
so we have plus 81.
And once again, we have two like terms, two X terms here,
9X plus 9X is equal to 18X,
so we have X squared plus 18X plus 81
as our simplified product.
And just remember, when squaring a binomial,
there are no shortcuts.
You need to write out both factors
and multiply like we normally would.
I hope you found this helpful.