Equation of a Normal Line - Algebraic Functions

$\mathit{y}\mathbf{=}\frac{{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{4}}{\mathit{x}\mathbf{+}\mathbf{1}}\mathbf{ }\mathbf{a}\mathbf{t}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{p}\mathbf{o}\mathbf{i}\mathbf{n}\mathbf{t}\mathbf{ }\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{ }\mathbf{-}\mathbf{4}\mathbf{)}\mathbf{ }$

$\mathbf{S}\mathbf{t}\mathbf{e}\mathbf{p}\mathbf{ }\mathbf{1}\mathbf{:}\mathbf{ }\mathbf{G}\mathbf{e}\mathbf{t}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{D}\mathbf{e}\mathbf{r}\mathbf{i}\mathbf{v}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{v}\mathbf{e}\mathbf{ }\mathbf{b}\mathbf{y}\mathbf{ }\mathbf{A}\mathbf{p}\mathbf{p}\mathbf{l}\mathbf{y}\mathbf{i}\mathbf{n}\mathbf{g}\mathbf{ }\mathbf{Q}\mathbf{u}\mathbf{o}\mathbf{t}\mathbf{i}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{ }\mathbf{R}\mathbf{u}\mathbf{l}\mathbf{e}$

$$\begin{gathered} \text{ \\ ddx(uv)=vdudx-udvdxv2} \end{gathered}$$

$\mathrm{Given}:$

$$\begin{gathered} \text{ \\ u=x2-4} \end{gathered}$$

$$\begin{gathered} \text{ \\ dudx=2x} \end{gathered}$$

$$\begin{gathered} \text{ \\ v=x+1} \end{gathered}$$

$$\begin{gathered} \text{ \\ dvdx=1 \\ } \end{gathered}$$

$\frac{dy}{\mathit{dx}}\mathit{=}\frac{x\mathit{+}\mathit{1}\mathit{\left(}\mathit{2}x\mathit{\right)}\mathit{-}{x}^{\mathit{2}}\mathit{-}\mathit{4}\mathit{\left(}\mathit{1}\mathit{\right)}}{\mathit{(}x\mathit{+}\mathit{1}{\mathit{)}}^{\mathit{2}}}$

$$\begin{gathered} \text{ \\ dydx=2x2+2x-x2-4(x+1)2} \end{gathered}$$

$$\begin{gathered} \text{ \\ dydx=x2+2x+4(x+1)2 \\ } \end{gathered}$$

$\mathbf{S}\mathbf{t}\mathbf{e}\mathbf{p}\mathbf{ }\mathbf{2}\mathbf{:}\mathbf{ }\mathbf{G}\mathbf{e}\mathbf{t}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{S}\mathbf{l}\mathbf{o}\mathbf{p}\mathbf{e}\mathbf{ }\mathbf{a}\mathbf{t}\mathbf{ }\mathbf{p}\mathbf{o}\mathbf{i}\mathbf{n}\mathbf{t}\mathbf{ }\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{ }\mathbf{-}\mathbf{4}\mathbf{)}$

$$\begin{gathered} \text{ \\ m=02+2(0)+4(0+1)2} \end{gathered}$$

$$\begin{gathered} \text{ \\ m=4 \\ } \end{gathered}$$

$\mathbf{S}\mathbf{t}\mathbf{e}\mathbf{p}\mathbf{ }\mathbf{3}\mathbf{:}\mathbf{ }\mathbf{A}\mathbf{p}\mathbf{p}\mathbf{l}\mathbf{y}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{e}\mathbf{q}\mathbf{u}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{ }\mathbf{o}\mathbf{f}\mathbf{ }\mathbf{a}\mathbf{ }\mathbf{N}\mathbf{o}\mathbf{r}\mathbf{m}\mathbf{a}\mathbf{l}\mathbf{ }\mathbf{L}\mathbf{i}\mathbf{n}\mathbf{e} \mathrm{at} \mathrm{point} (0, -4)$

$$\begin{gathered} \text{ \\ y-y1=-1m(x-x1)} \end{gathered}$$

$$\begin{gathered} \text{ \\ y+4=-14(x-0)-4} \end{gathered}$$

$$\begin{gathered} \text{ \\ -4y-16=x} \end{gathered}$$

$$\begin{gathered} \text{ \\ x+4y+16=0 \\ } \end{gathered}$$

$\mathbf{F}\mathbf{i}\mathbf{n}\mathbf{a}\mathbf{l}\mathbf{ }\mathbf{A}\mathbf{n}\mathbf{s}\mathbf{w}\mathbf{e}\mathbf{r}\mathbf{:}\mathit{ }\mathit{x}\mathit{+}\mathit{4}\mathit{y}\mathit{+}\mathit{16}\mathit{=}\mathit{0}$