To multiply the given polynomials, [tex]\((5x^2 - 7)\)[/tex] and [tex]\((5x^2 + 7)\)[/tex], we will use the distributive property (also known as the FOIL method for binomials). Let's go through this step-by-step:
1. First, multiply the first terms of each binomial:
[tex]\[
(5x^2) \cdot (5x^2) = 25x^4
\][/tex]
2. Outer, multiply the outer terms of the binomials:
[tex]\[
(5x^2) \cdot 7 = 35x^2
\][/tex]
3. Inner, multiply the inner terms of the binomials:
[tex]\[
(-7) \cdot (5x^2) = -35x^2
\][/tex]
4. Last, multiply the last terms of each binomial:
[tex]\[
(-7) \cdot 7 = -49
\][/tex]
Now, add all the products together:
[tex]\[
25x^4 + 35x^2 - 35x^2 - 49
\][/tex]
Notice that the middle terms [tex]\(+35x^2\)[/tex] and [tex]\(-35x^2\)[/tex] cancel each other out, leaving us with:
[tex]\[
25x^4 - 49
\][/tex]
So, the product of the two polynomials [tex]\((5x^2 - 7)\)[/tex] and [tex]\((5x^2 + 7)\)[/tex] is:
[tex]\[
25x^4 - 49
\][/tex]
Thus, when you arrange the answer in descending powers of [tex]\(x\)[/tex], you get:
[tex]\[
\boxed{25x^4 - 49}
\][/tex]
