5sin^2x+3sinxcosx-2cos^2x=3

5sin^2(x) + 3sin(x)cos(x) - 2cos^2(x) = 3;
5sin^2(x) + 3sin(x)cos(x) - 2cos^2(x) - 3cos^2(x) - 3sin^2(x) = 0;
2sin^2(x) + 3sin(x)cos(x) - 5cos^2(x) = 0;
Делим на cos^2(x) ≠ 0;
2tg^2(x) + 3tg(x) - 5 = 0;
tg(x) = y - замена переменной;
2y^2 + 3y - 5 = 0;
D = 9 + 40 = 49;
y1 = (-3 + 7) / 4 = 1;
y2 = (- 3 - 7) / 4 = - 5/2;
1) tg(x) = 1; x = п/4 + пn, n∈Z.
2) tg(x) = -5/2; x = -arctg(5/2) + пk, k∈Z.
Ответ: x = п/4 + пn, n∈Z; x = -arctg(5/2) + пk, k∈Z.