To multiply and simplify the expression [tex]\((4x^2 - 1)(6x^2 - 5)\)[/tex], we can use the distributive property (also known as the FOIL method for binomials). Let's go through it step by step.
First, we will distribute each term in the first polynomial to each term in the second polynomial.
1. Multiply [tex]\(4x^2\)[/tex] by [tex]\(6x^2\)[/tex]:
[tex]\[
4x^2 \cdot 6x^2 = 24x^4
\][/tex]
2. Multiply [tex]\(4x^2\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
4x^2 \cdot (-5) = -20x^2
\][/tex]
3. Multiply [tex]\(-1\)[/tex] by [tex]\(6x^2\)[/tex]:
[tex]\[
-1 \cdot 6x^2 = -6x^2
\][/tex]
4. Multiply [tex]\(-1\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
-1 \cdot (-5) = 5
\][/tex]
Next, we add all these results together:
[tex]\[
24x^4 - 20x^2 - 6x^2 + 5
\][/tex]
Combine the like terms ([tex]\(-20x^2\)[/tex] and [tex]\(-6x^2\)[/tex]):
[tex]\[
24x^4 - 26x^2 + 5
\][/tex]
Thus, the simplified form of the expression is:
[tex]\[
24x^4 - 26x^2 + 5
\][/tex]
So, the correct answer is:
(D) [tex]\(24 x^4 - 26 x^2 + 5\)[/tex]
