Certainly! Let's expand and simplify the given expression step-by-step. The expression we need to expand is:
[tex]\[
\left(y^2 - 9\right)\left(y^2 - 4\right)
\][/tex]
First, let's recall the distributive property, which states that for any expressions [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex]:
[tex]\[
(a + b)(c + d) = ac + ad + bc + bd
\][/tex]
For our expression, we will expand each term in the first binomial [tex]\((y^2 - 9)\)[/tex] by each term in the second binomial [tex]\((y^2 - 4)\)[/tex]:
[tex]\[
(y^2 - 9)(y^2 - 4)
\][/tex]
1. First Multiply [tex]\(y^2\)[/tex] with each term in [tex]\((y^2 - 4)\)[/tex]:
[tex]\[
y^2 \cdot y^2 = y^4
\][/tex]
[tex]\[
y^2 \cdot (-4) = -4y^2
\][/tex]
2. Next, Multiply [tex]\(-9\)[/tex] with each term in [tex]\((y^2 - 4)\)[/tex]:
[tex]\[
-9 \cdot y^2 = -9y^2
\][/tex]
[tex]\[
-9 \cdot (-4) = 36
\][/tex]
Now, combine all these terms together:
[tex]\[
y^4 - 4y^2 - 9y^2 + 36
\][/tex]
3. Combine like terms:
[tex]\[
-4y^2 - 9y^2 = -13y^2
\][/tex]
Putting it all together in standard polynomial form:
[tex]\[
y^4 - 13y^2 + 36
\][/tex]
So, the expanded form of [tex]\(\left(y^2 - 9\right)\left(y^2 - 4\right)\)[/tex] is:
[tex]\[
y^4 - 13y^2 + 36
\][/tex]
Thus, the final answer is:
[tex]\[
\left(y^2 - 9\right)\left(y^2 - 4\right) = y^4 - 13y^2 + 36
\][/tex]
