4sin²x - 3 sinx * cosx + 5 cos² x = 3

4sin^2 (x) - 3sin (x) cos (x) + 5 cos^2 (x) = 3;

4sin^2 (x) - 3sin (x) cos (x) + 5 cos^2 (x) - 3 = 0;

4sin^2 (x) - 3sin (x) cos (x) + 5 cos^2 (x) - 3sin^2 (x) - 3cos^2 (x) = 0;

sin^2 (x) - 3sin (x) cos (x) + 2 cos^2 (x) = 0;

Разделим обе части на cos^2 (x) ≠ 0;

tg^2 (x) - 3tg (x) + 2 = 0;

Замена переменной tg (x) = y;

y^2 - 3y + 2 = 0;

D = 1;

y1 = 1;

y2 = 2;

Обратная замена переменной:

tg (x) = 1; = => x = п/4 + пn, n∈Z;

tg (x) = 2; = => x = arctg (2) + пk, k∈Z.

Ответ: x = п/4 + пn, n∈Z; x = arctg (2) + пk, k∈Z.