Understanding the Context

Consider the function defined by:
$$
f(t) = 2(t + 3)^2 - 12(t + 3) + 13
$$
Here, the variable $ t $ is defined as $ t = x^2 - 3 $, so $ t + 3 = x^2 $. This substitution reveals a key simplification: every occurrence of $ t + 3 $ corresponds directly to $ x^2 $, enabling us to rewrite $ f(t) $ entirely in terms of $ x $.

Step-by-Step Expansion

Begin by substituting $ t + 3 = x^2 $ directly into $ f(t) $:
$$
f(x^2) = 2(x^2)^2 - 12(x^2) + 13 = 2x^4 - 12x^2 + 13
$$

However, to demonstrate full expansion, let us expand the original expression fully in terms of $ t $:
$$
f(t) = 2(t + 3)^2 - 12(t + 3) + 13
$$

Key Insights

First, expand $ (t + 3)^2 = t^2 + 6t + 9 $, so:
$$
2(t^2 + 6t + 9) = 2t^2 + 12t + 18
$$

Next, expand $ -12(t + 3) = -12t - 36 $

Combine all terms:
$$
2t^2 + 12t + 18 - 12t - 36 + 13 = 2t^2 + (12t - 12t) + (18 - 36 + 13) = 2t^2 - 5
$$

Thus, we find:
$$
f(t) = 2t^2 - 5
$$

Function Transformation: $ f(x^2 + 1) $

Final Thoughts

Now leverage the simplified form to compute $ f(x^2 + 1) $. Replace $ t $ with $ x^2 + 1 $:
$$
f(x^2 + 1) = 2(x^2 + 1)^2 - 5
$$

Expand $ (x^2 + 1)^2 = x^4 + 2x^2 + 1 $, so:
$$
2(x^4 + 2x^2 + 1) - 5 = 2x^4 + 4x^2 + 2 - 5 = 2x^4 + 4x^2 - 3
$$

Final Result

The fully simplified expression is:
$$
oxed{2x^4 + 4x^2 - 3}
$$

Why This Technique Matters

This substitution-based approach is valuable in both academic problem-solving and real-world modeling. By recognizing shared structures through variables like $ t + 3 $, and expanding methodically, complex transformations become systematic and error-free. Whether simplifying polynomial functions or solving recursive relations, mastering substitution greatly enhances algebraic fluency.