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.