Multiply the binomials.

[tex]
\begin{array}{c}
\left(7x^2 - 3y^2\right)\left(x^2 - 8y^2\right) \\
\left(7x^2 - 3y^2\right)\left(x^2 - 8y^2\right) = 7x^4 - 59x^2y^2 + 24y^4
\end{array}
[/tex]

(Simplify your answer.)



Answer :

Sure! Let's work through the detailed steps of multiplying the binomials [tex]\((7x^2 - 3y^2)(x^2 - 8y^2)\)[/tex] and simplifying the answer.

1. Distribute each term in the first binomial to each term in the second binomial:

[tex]\[ (7x^2 - 3y^2)(x^2 - 8y^2) = 7x^2 \cdot x^2 + 7x^2 \cdot (-8y^2) + (-3y^2) \cdot x^2 + (-3y^2) \cdot (-8y^2) \][/tex]

2. Calculate each product:

- [tex]\(7x^2 \cdot x^2 = 7x^4\)[/tex]
- [tex]\(7x^2 \cdot (-8y^2) = -56x^2y^2\)[/tex]
- [tex]\((-3y^2) \cdot x^2 = -3x^2y^2\)[/tex]
- [tex]\((-3y^2) \cdot (-8y^2) = 24y^4\)[/tex]

3. Combine like terms (terms with the same variables and exponents):

Combining [tex]\(-56x^2y^2\)[/tex] and [tex]\(-3x^2y^2\)[/tex]:

[tex]\[ -56x^2y^2 + (-3x^2y^2) = -59x^2y^2 \][/tex]

4. Write out the simplified expression by combining all of the products:

[tex]\[ 7x^4 - 59x^2y^2 + 24y^4 \][/tex]

Therefore, the expanded and simplified form of the product of the binomials [tex]\((7x^2 - 3y^2)(x^2 - 8y^2)\)[/tex] is:

[tex]\[ 7x^4 - 59x^2y^2 + 24y^4 \][/tex]