Question

1. factor (75a^4 - 3):

a. (3(5a^2 + 1)(5a^2 - 1))
b. ((5a^2 - 1)^2)
c. (3(25a^2 + 1)^2)
d. (9(5a^2 - 1)^2)

Explanation:

Step1: Factor out GCF

Identify greatest common factor of $75a^4$ and $3$, which is 3.
$75a^4 - 3 = 3(25a^4 - 1)$

Step2: Apply difference of squares

Recognize $25a^4 -1$ as $(5a^2)^2 - 1^2$, use $x^2-y^2=(x+y)(x-y)$.
$25a^4 - 1 = (5a^2 + 1)(5a^2 - 1)$

Step3: Combine results

Substitute factored form back into the expression.
$3(25a^4 - 1) = 3(5a^2 + 1)(5a^2 - 1)$

Answer:

a. $3(5a^2 + 1)(5a^2 - 1)$