Ellipse $4x^2+y^2+ax+by+c=0$ passes through the point $(-1,2)$ and is tangent to the x-axis 
Question:
Find the values of the constants $a$,$b$ and $c$ such that the ellipse $4x^2+y^2+ax+by+c=0$ passes through the point $(-1,2)$ and is tangent to the x-axis

What I've done so far:
Substitute $(-1,2)$ in $4x^2+y^2+ax+by+c=0$: 
$$\begin{align}
&\implies 4+4-a+2b+c=0 \\
&\implies c=a-2b-8 \tag{1}
\end{align}$$
Rewriting the equation of the ellipse to get standard form:
$$\begin{align}
& 4x^2+y^2+ax+by=-c \\[6pt]
\implies& 4\left(x^2+\frac{a}{4}x+\frac{a^2}{64}\right)
+\left(y^2+by+\frac{b^2}{4}\right)=a-2b-8+\frac{a^2}{16}+\frac{b^2}{4} \\[6pt]
\implies& \frac{4\left(x+\frac{a}{8}\right)^2}{a-2b-8+\frac{a^2}{16}+\frac{b^2}{4}}+\frac{\left(y+\frac{b}{2}\right)^2}{a-2b-8+\frac{a^2}{16}+\frac{b^2}{4}}=1
\end{align}$$
Since the ellipse is tangent to the $x$-axis:
$$\begin{align}
&\quad\sqrt{a-2b-8+\frac{a^2}{16}+\frac{b^2}{4}} = \frac{b}{2} \\[6pt]
\implies&\quad a-2b-8+\frac{a^2}{16}+\frac{b^2}{4} = \frac{b^2}{4} \\
\implies&\quad 2b= a-\frac{a^2}{16}-8 \tag{2}
\end{align}$$
From $(1)$ and $(2)$:
$$\quad c= a-\left(a-\frac{a^2}{16}-8\right)- 8 \quad\implies\quad c= \frac{a^2}{16} \tag{3}$$ 
Therefore, the equation of the ellipse becomes
$$\frac{4\left(x+\frac{a}{8}\right)^2}{a-(a-\frac{a^2}{16}-8)-8+\frac{a^2}{16}+\frac{b^2}{4}}+\frac{(y+\frac{b}{2})^2}{a-(a-\frac{a^2}{16}-8)-8+\frac{a^2}{16}+\frac{b^2}{4}}=1$$ 
That is,
$$\frac{\left(x+\frac{a}{8}\right)^2}{b^2/16}+\frac{\left(y+\frac{b}{2}\right)^2}{b^2/4}=1 \tag{4}$$ 

I'm stuck at this point and don't know how to find $a$ (or what else to do in general).
All help is appreciated.

Your equation $(1)$ is correct:
$$a - 2 b - c = 8 \tag{1}$$
For the tangency condition, you're thinking a little too hard. If the ellipse is tangent to the $x$-axis, then the equation should have a double-root for $x$ when $y=0$; that is, its discriminant must vanish:
$$4 x^2 + a x + c = 0 \quad\to\quad \text{discriminant} = a^2 - 4\cdot 4\cdot c = 0 \quad\to\quad a^2 = 16 c \tag{2}$$
... which is, of course, your $(3)$. Now, two independent equations aren't enough to specify three unknowns. There are lots of solutions here. Since $a^2 \geq 0$, we know $c$ is non-negative; take $c := (2d)^2$ for some $d$ (the factor of $2$ helps us avoid some fractions), and we have

$$a = \pm 8 d \qquad b = -2( d^2 \mp 2 d + 2 ) \qquad c = 4d^2 \tag{3}$$

For completeness, the equation of the ellipse is ...
$$4 \left(x \pm d \right)^2 + \left( y -(d^2\mp 2 d+2) \right)^2 = (d^2\mp 2 d + 2)^2 \tag{4}$$
Note that the vertical radius is $\left|d^2\mp 2 d + 2\right|$, which matches the vertical offset of the center, easily re-confirming the tangency property. $\square$
