Solve for x: 2sin^2x+5sinxcosx+5cos^2x=1

2sin^2x+5sinxcosx
+5cos^2x=1.

5sinxcosx =
1-5cos^2x-2sin^2x.

5sinxcosx =
1-5(1-sin^2x)-2sin^2x.

5sinxcosx =
1-5+5sin^2x-2sin^2x

5sinxcosx =
3sin^2-4.

We square both
sides:

25sin^2xcos^2x =
(3sin^2x-4)^2

25sin^2x (1-sin^2x) =  9
sin^4x-24sin^2x+16

25sin^2-25sin^4 = 9sin^4
-24sin^2x+16.

9s^4+25s^4 - 49s^2 +16 = 0, where s=
sinx.

34s^4 -49s^2+16 =
0.

34s^4 - 17s^2 - 32s^2+16 =
0

17s^2(2s^2-1)-16(2s^2-1) =
0

(2s^2-1((17s^2-16) = 0

s^2=
1/2, Or s^2 = 16/17

 s= +sqrt1/2  , sinx = sqrt(1/2), or x
= 45 or  135 deg

s= -sqrt(1/2)., sinx = -sqrt(1/2), or x=
225, or x= 360-45 = 315 deg

s = sqrt(16/17) , sinx =
sqr(16/17) , x =  75.96 deg, or  104.04 deg

s =
-sqrt(16/17), sinx = -sqr(16/17), x =  -75.96 deg = 360-75.96 = 284.94 deg or 180+75.96
= 255.96 deg