3cos^2x+4sinxcosx+5sin^2x=2

$3cos ^{2} x+4sinxcosx+5sin ^{2} x=2$
$3cos ^{2} x+4sinxcosx+5sin ^{2} x-2(sin ^{2} x+cos ^{2} x)=0$
$3cos ^{2} x+4sinxcosx+5sin ^{2} x-2sin ^{2} x-2cos ^{2} x=0$
$cos ^{2} x+4sinxcosx+3sin ^{2} x=0$ |: $cos ^{2} x,$ $(cos ^{2} x \neq 0)$

$1+4 \frac{sinx}{cosx} +3 \frac{sin^{2}x }{cos ^{2}x } =0$

$1+4tgx+3tg ^{2} x=0$
$tgx=y$
$3y ^{2} +4y+1=0$
D = 4² - 4· 3·1 = 16 - 12 = 4
$\sqrt{D} =4$

$y _{1} = \frac{-4+2}{6} = \frac{-2}{6} =- \frac{1}{3}$

$y _{2} = \frac{-4-2}{6} = \frac{-6}{6} =-1$

$\left \{ {{tgx=- \frac{1}{3} } \atop {tgx=-1}} \right.$

$x=-arctg \frac{1}{3} + \pi n,$ n∈Z
$x=- \frac{ \pi }{4} + \pi m,$ m∈Z