<s>[INST] Find the range of $f(x)=11\cos^2x+3\sin^2x+6\sin x\cos x+5$ I'm trying to solve this problem.

Find the range of $f(x)=11\cos^2x+3\sin^2x+6\sin x\cos x+5$

I have simplified this problem to $$f(x)= 8\cos^2x+6\sin x\cos x+8$$ and tried working with $g(x)= 8\cos^2x+6\sin x\cos x$.
I factored out the $2\cos x$ and rewrote the other factor as a linear combination of cosine.
It reduces down to $$g(x)=10\cos x\cos\left(x-\tan^{-1}\frac{3}{4}\right)$$
But then I'm stuck here. Please help me. Perhaps there's a different way to approach this?
 [/INST] Hint:
Let $y=A\sin^2x+B\cos^2x+C\sin x\cos x+D$
Method$\#1:$ Divide both sides by $\cos^2x$  to form a Quadratic Equation in $\tan x$
As $\tan x$ is real, the discriminant must be $\ge0$
Method$\#2:$ Divide both sides by $\sin^2x$
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