To factor the polynomial [tex]\( x^4 + 12x^2 + 32 \)[/tex] completely, we can follow these steps:
1. Identify the Structure of the Polynomial:
Observe that the polynomial [tex]\( x^4 + 12x^2 + 32 \)[/tex] can be thought of as a quadratic in terms of [tex]\( x^2 \)[/tex]. That is, let [tex]\( y = x^2 \)[/tex]. This transforms the polynomial into:
[tex]\[
y^2 + 12y + 32
\][/tex]
2. Factor the Quadratic Expression:
Next, we need to factor [tex]\( y^2 + 12y + 32 \)[/tex] into two binomials. We look for two numbers that multiply to 32 and add up to 12. These numbers are 4 and 8. Hence, we can rewrite the quadratic as:
[tex]\[
y^2 + 12y + 32 = (y + 4)(y + 8)
\][/tex]
3. Substitute Back [tex]\( y = x^2 \)[/tex]:
Now, re-substitute [tex]\( y \)[/tex] with [tex]\( x^2 \)[/tex]:
[tex]\[
(y + 4)(y + 8) \rightarrow (x^2 + 4)(x^2 + 8)
\][/tex]
4. Conclusion:
Therefore, the completely factored form of the polynomial [tex]\( x^4 + 12x^2 + 32 \)[/tex] is:
[tex]\[
(x^2 + 4)(x^2 + 8)
\][/tex]
5. Select the Correct Answer:
Among the given options, the correct one is:
[tex]\[
\boxed{(x^2 + 4)(x^2 + 8)}
\][/tex]
