Answer :
To factor the expression [tex]\(45x^6 - 25x^2 - 25\)[/tex], we can look for a common factor among the terms. In this case, the common factor is 5.
So, we can rewrite the expression as:
[tex]\[5(9x^6 - 5x^2 - 5)\][/tex]
Now, let's focus on factoring the trinomial[tex]\(9x^6 - 5x^2 - 5\)[/tex].
This trinomial doesn't seem to factor further using common factoring techniques. However, we can try to rewrite it as a difference of squares. Notice that[tex]\(9x^6 - 5x^2 - 5\) can be written as \(9x^6 - (5x^2 + 5)\)[/tex].
We can rewrite [tex]\(5x^2 + 5\) as \(5(x^2 + 1)\)[/tex], which resembles a difference of squares if we consider[tex]\(x^2\) as \(x^2\) and \(1\) as \(1^2\).[/tex]
So, our expression becomes:
[tex]\[5(9x^6 - (5x^2 + 5))\]\[5(9x^6 - 5(x^2 + 1))\][/tex]
Now, we have a difference of squares inside the parentheses:
[tex]\[5\left((3x^3)^2 - (5)^2\right)\][/tex]
Using the difference of squares formula[tex]\(a^2 - b^2 = (a + b)(a - b)\)[/tex], where [tex]\(a = 3x^3\) and \(b = 5\),[/tex] we get:
[tex]\[5\left( (3x^3 + 5)(3x^3 - 5) \right)\][/tex]
Thus, the factored form of the expression [tex]\(45x^6 - 25x^2 - 25\) is \(5(3x^3 + 5)(3x^3 - 5)\).[/tex]
