Answer:
[tex](x + 1)(x - 1)(x^2 + 7)[/tex]
Step-by-step explanation:
To factor the expression [tex](x^2 + 5)^2 - 4(x^2 + 5) -12[/tex]
First, let y = (x² + 5) to make the expression easier
Then the expression becomes
y² - 4y -12
We can factor this as follows by noting that -6 x 2 = -12 and -6 + 2 = -4
y² - 4y -12 = y² - 6y + 2y -12
= y(y-6) +2(y - 6)
Taking out the common term y - 6, we get
(y - 6)(y + 2)
Substituting back x² + 5 for y we get
(x² + 5 - 6)(x² + 5 + 2)
= (x² - 1)(x² + 7)
We can factor x² - 1 further as follows using the rule(a + b)(a - b) = a² - b² giving
x² - 1 = (x + 1)(x-1)
So the complete factorization of
[tex](x^2 + 5)^2 - 4(x^2 + 5) -12 = (x + 1)(x - 1)(x^2 + 7)[/tex]
