Solve the Fraction Equation: Decompose (x+6) = (x^2+6)/(x-4)

Let's solve the given equation:

$x+6=\frac{x^2+6}{x-4}$

We'll start by defining the domain of the unknown in the equation, this is because there is a variable in the denominator on the right side, we'll remember that the denominator cannot be zero and therefore we require that the expression in the denominator is not zero, meaning:

$x-4\neq0$

Next we'll solve the (point) inequality identically to how we solve a regular equation:

$x-4\neq0\\ \downarrow\\ \boxed{x\neq4}$

Therefore the domain of the unknown in the equation is:

$\textcolor{red}{\boxed{x\neq4}}$

(Note that any point inequality (and not slope inequality, meaning where $\neq$ and not \leq,\geq,>,< ) is solved identically to a regular equation)

Let's return to the given equation:

$x+6=\frac{x^2+6}{x-4}$

First we'll represent each term in the equation that isn't a fraction as a fraction, we'll do this using the fact that dividing any number by 1 doesn't change its value:

$x+6=\frac{x^2+6}{x-4} \\ \downarrow\\ \frac{ x}{1}+ \frac{6}{1}=\frac{x^2+6}{x-4}$

Next - we want to eliminate the fraction line, we'll do this by multiplying both sides of the equation by the simplest common denominator, in this case the simplest common denominator is the simple expression: $x-4$, this is because the other denominators are 1, therefore we'll multiply both sides of the equation by this expression, while remembering that the expressions in the numerator of each of the fractions in the equation will be multiplied by the expression that answers the question: "By how much did we multiply the current denominator to get the common denominator?" (for each fraction separately - see in the next calculation), then we'll open parentheses and simplify the equation by moving terms and combining like terms:

$\frac{x^{\diagdown\cdot(x-4)}}{1}+ \frac{6^{\diagdown\cdot(x-4)}}{1}=\frac{x^2+6^{\diagdown\cdot1}}{x-4} \text{/}\cdot(x-4) \\ \downarrow\\ x\cdot(x-4)+6\cdot(x-4)=(x^2+6)\cdot1\\ x^2-4x+6x-24=x^2+6 \\ 2x=30$

Let's continue, we got a first-degree equation, let's solve it in the regular way:

$2x=30 \hspace{6pt}\text{/}:2\\ \boxed{x=15}$

Now let's not forget the domain of the unknown in the equation:

$\boxed{x\neq4}$

Note that the solution we found does not contradict the domain and therefore the final solution is:

$\boxed{x=15}$

Let's summarize the solution of the equation:

$x+6=\frac{x^2+6}{x-4}\textcolor{red}{\rightarrow \boxed{x\neq4}} \\ \downarrow\\ \frac{ x}{1}+ \frac{6}{1}=\frac{x^2+6}{x-4}\\ \frac{x^{\diagdown\cdot(x-4)}}{1}+ \frac{6^{\diagdown\cdot(x-4)}}{1}=\frac{x^2+6^{\diagdown\cdot1}}{x-4} \text{/}\cdot(x-4) \\ \downarrow\\ x\cdot(x-4)+6\cdot(x-4)=(x^2+6)\cdot1\\ 2x=30 \rightarrow\boxed{x=15}\\ \downarrow\\ x=15\stackrel{?}{\textcolor{red}{\leftrightarrow} }\textcolor{red}{x\neq4}\\ \downarrow\\ \boxed{x\textcolor{green}{\stackrel{!}{=}}15}$

Therefore the correct answer is answer D.