rewrite <- le_plus_minus; try (simple apply lt_le_weak; auto with arith).
replace (3 * q) with (2 * q + q); try ring.
apply plus_lt_le_compat; auto.
repeat rewrite (mult_comm 2); apply mult_lt; auto with arith.
Qed.
Remark mult_minus_distr_l : forall a b c : nat, a * (b - c) = a * b - a * c.
intros a b c; repeat rewrite (mult_comm a); apply mult_minus_distr_r.
Qed.
Remark minus_eq_decompose :
forall a b c d : nat, a = b -> c = d -> a - c = b - d.
intros a b c d H H0; rewrite H; rewrite H0; auto.
Qed.

Theorem new_equality :
(3 * p - 4 * q) * (3 * p - 4 * q) = 2 * ((3 * q - 2 * p) * (3 * q - 2 * p)).