To factor the expression [tex]\( 40x^2 - 10 \)[/tex], follow these steps:
1. First, look for any common factors in the terms of the expression. Here, both terms [tex]\( 40x^2 \)[/tex] and [tex]\( 10 \)[/tex] have a common factor of 10. So, we can factor out 10:
[tex]\[
40x^2 - 10 = 10(4x^2 - 1)
\][/tex]
2. Next, observe that [tex]\( 4x^2 - 1 \)[/tex] is a difference of squares. We know the formula for the difference of squares is:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
3. Rewrite [tex]\( 4x^2 - 1 \)[/tex] in the form [tex]\( a^2 - b^2 \)[/tex]:
[tex]\[
4x^2 - 1 = (2x)^2 - 1^2
\][/tex]
4. Apply the difference of squares formula:
[tex]\[
(2x)^2 - 1^2 = (2x - 1)(2x + 1)
\][/tex]
5. Substitute back into the expression where we factored out the 10:
[tex]\[
40x^2 - 10 = 10(2x - 1)(2x + 1)
\][/tex]
Therefore, the factored form of the expression [tex]\( 40x^2 - 10 \)[/tex] is:
[tex]\[
10(2x - 1)(2x + 1)
\][/tex]
Since none of the provided answer choices match exactly with our factored form at first glance, let's ensure there isn't simplification or a different representation that matches. After confirming, the correct answer is:
[tex]\(\boxed{None of the above}\)[/tex]
(Note: Given the provided choices A, B, C, and D, and verifying the detailed factorization steps, none of these match the correct factorization of [tex]\( 40x^2 - 10 \)[/tex]. If this is not the expected final answer, there might be an error or need for reconsideration in the provided options.)
