Sure, let's factor the given expression step-by-step.
We start with the polynomial:
[tex]\[
x^4 - 7x^2 + 12
\][/tex]
Our goal is to factor this polynomial. Notice that [tex]\( x^4 - 7x^2 + 12 \)[/tex] is a quadratic in form when substituting [tex]\( y = x^2 \)[/tex]. This transforms the polynomial into:
[tex]\[
y^2 - 7y + 12
\][/tex]
We need to factor this quadratic expression. So, we look for two numbers that multiply to 12 (the constant term) and add up to -7 (the coefficient of the linear term). These numbers are -3 and -4. Therefore, we can factor [tex]\( y^2 - 7y + 12 \)[/tex] as:
[tex]\[
(y - 3)(y - 4)
\][/tex]
Re-substituting [tex]\( y = x^2 \)[/tex] back in, we get:
[tex]\[
(x^2 - 3)(x^2 - 4)
\][/tex]
Now, we notice that [tex]\( x^2 - 4 \)[/tex] is a difference of squares, which can be further factored:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
Thus, the entire factorization of [tex]\( x^4 - 7x^2 + 12 \)[/tex] is:
[tex]\[
(x - 2)(x + 2)(x^2 - 3)
\][/tex]
This matches the format [tex]\((x^2 - 4)(x^2 \text{ [?] } 3)\)[/tex]. To fill in the [?], we see that the factor corresponding to [tex]\( x^2 \)[/tex] and 3 is [tex]\( (x^2 - 3) \)[/tex], which uses the minus symbol.
So, the correct symbol to use is:
[tex]\[
\boxed{-}
\][/tex]
