To solve the equation [tex]\(16(x^3 + 1)^2 - 22(x^3 + 1) - 3 = 0\)[/tex] and rewrite it as a quadratic equation, we can use an appropriate substitution.
Given the equation [tex]\(16(x^3 + 1)^2 - 22(x^3 + 1) - 3 = 0\)[/tex]:
1. We notice that the expression [tex]\(x^3 + 1\)[/tex] appears repeatedly. This suggests a substitution that can simplify it.
2. Let's denote [tex]\(u = x^3 + 1\)[/tex]. By making this substitution, the equation then simplifies into terms involving [tex]\(u\)[/tex] only.
3. Substituting [tex]\(u = x^3 + 1\)[/tex] into the equation, we have:
[tex]\[
16(u)^2 - 22(u) - 3 = 0
\][/tex]
4. This is now a quadratic equation in terms of [tex]\(u\)[/tex], specifically:
[tex]\[
16u^2 - 22u - 3 = 0
\][/tex]
Thus, the correct substitution to use is [tex]\(u = x^3 + 1\)[/tex].
Therefore, the answer is:
[tex]\[
\boxed{u = (x^3 + 1)}
\][/tex]
