Sure, let's multiply the given binomials step by step:
Given binomials:
[tex]\[
(7x^2 - 3y^2)(x^2 - 8y^2)
\][/tex]
To simplify, we will use the distributive property, also known as the FOIL method for binomials:
1. First terms: Multiply the first term of each binomial:
[tex]\[
7x^2 \cdot x^2 = 7x^4
\][/tex]
2. Outer terms: Multiply the outer terms:
[tex]\[
7x^2 \cdot (-8y^2) = -56x^2y^2
\][/tex]
3. Inner terms: Multiply the inner terms:
[tex]\[
(-3y^2) \cdot x^2 = -3x^2y^2
\][/tex]
4. Last terms: Multiply the last term of each binomial:
[tex]\[
(-3y^2) \cdot (-8y^2) = 24y^4
\][/tex]
Now, add all these products together:
[tex]\[
7x^4 + (-56x^2y^2) + (-3x^2y^2) + 24y^4
\][/tex]
Combine like terms (the [tex]\(x^2y^2\)[/tex] terms):
[tex]\[
7x^4 - 56x^2y^2 - 3x^2y^2 + 24y^4 = 7x^4 - 59x^2y^2 + 24y^4
\][/tex]
Therefore, the product of the given binomials is:
[tex]\[
7x^4 - 59x^2y^2 + 24y^4
\][/tex]
