= Solution
Let
$$
S=\sum_{T<t\leq2T}\sum_{R<r\leq2R}b_tc_r e(\alpha r^2t^2).
$$
Applying the <Cauchy-Schwarz inequality> in $t$ gives
$$
|S|^2
\leq\|b\|_2^2
\sum_t\left|\sum_r c_re(\alpha r^2t^2)\right|^2
=\|b\|_2^2
\sum_{r_1,r_2}c_{r_1}\overline{c_{r_2}}
\sum_t e\bigl(\alpha t^2(r_1^2-r_2^2)\bigr).
$$
Take absolute values and apply Cauchy-Schwarz to $(r_1,r_2)$. Since $\sum_{r_1,r_2}|c_{r_1}c_{r_2}|^2=\|c\|_2^4$, expanding the remaining square gives
$$
|S|^4
\leq\|b\|_2^4\|c\|_2^4
\sum_{\substack{T<t_1,t_2\leq2T\\R<r_1,r_2\leq2R}}
e\bigl(\alpha(t_1^2-t_2^2)(r_1^2-r_2^2)\bigr).
$$
Taking fourth roots proves the claimed <bilinear quadratic exponential sum fourth-moment bound>.
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