Multiply the following polynomials, then place the answer in the proper location on the grid. Write the answer in descending powers of [tex]$x$[/tex].

[tex]\left(5x^2 - 7\right)\left(5x^2 + 7\right)[/tex]



Answer :

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]