Solution: Let $ t = x^2 - 3 $, so $ x^2 = t + 3 $. Substitute into the equation: $ f(t) = 2(t + 3)^2 - 12(t + 3) + 13 $. Expanding: $ 2(t^2 + 6t + 9) - 12t - 36 + 13 = 2t^2 + 12t + 18 - 12t - 36 + 13 = 2t^2 - 5 $. Thus, $ f(x^2 + 1) = 2(x^2 + 1)^2 - 5 = 2x^4 + 4x^2 + 2 - 5 = 2x^4 + 4x^2 - 3 $. oxed{2x^4 + 4x^2 - 3}

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# Mastering Algebraic Substitution: Simplifying Complex Functions Solving mathematical expressions often requires clever substitutions and strategic expansion to simplify complicated forms into more manageable ones. One effective algebraic technique involves analyzing a function expressed in terms of a shifted variable, substituting strategically, and expanding carefully. In this article, we explore a powerful method illustrated through a clear substitution example that transforms a somewhat complex function into a simplified polynomial. ## The Substitution Strategy 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 $.