Domination of quadratic plus linear terms by a quadratic bound

For all non-negative coefficients $c_1, c_2 \in \mathbb{N}$ and any length $n \in \mathbb{N}$, the sum of a quadratic term and a linear term is dominated by a pure quadratic term with summed coefficients:

Expanding the right-hand side via distributivity yields $c_1 (n + 1)^2 + c_2 (n + 1)^2$. The inequality then reduces to $c_2 (n + 1) \le c_2 (n + 1)^2$. Since $n + 1 \ge 1$, we have $n + 1 \le (n + 1)^2$, and multiplying by $c_2$ preserves the inequality.

This algebraic bound allows bounding the composite runtime of a two-stage Turing machine (quadratic transformation followed by linear sweep) by a single quadratic cost.