If $x$ and $y$ are real numbers such that $4x^2 + y^2 = 4x - 2y + 7$, find the maximum value of $5x+6y$. If $x$ and $y$ are real numbers such that $4x^2 + y^2 = 4x - 2y + 7$, find the maximum value of $5x+6y$.
I did a little bit of manipulation and got $4(x+1)(x-2) + (y+1)^2=0$. I then got $x=2$ and $y=-1$ which means the maximum must be $4$, but the answer key says it's $16$. How come?

This can be solved as a tangency problem.
Given the non degenerated conic $C(x,y) = 0$ determine the value for $\lambda$ such that $a x + b y = \lambda$ is tangent to $C(x,y)=0$ so we proceed as follows:
first we substitute $y = \frac 1b(\lambda-a x)$ into $C(x,y) = 0$ giving the relationship
$$
C\left(x, \frac 1b(\lambda-a x)\right)=0
$$
second, we solve for $x$ giving the condition
$$
x^* = \frac{1}{169} \left(102\pm12 \sqrt{368-\lambda^2-7 \lambda}+5 \lambda\right)
$$
but at tangency $x^*$ is unique so follows the condition
$$
368-\lambda^2-7 \lambda = 0
$$
which solved gives
$$
\lambda^* = \{-23, 16\}
$$
hence the answer is $\lambda^* =16$
