Enter the values for the highlighted variables that show how to subtract the rational expressions correctly: \( \begin{aligned} \frac{2}{x^{2}-36}-\frac{1}{x^{2}+6 x} & =\frac{2}{(x+6)(x-6)}-\frac{1}{x(x+a)} \\ & =\frac{b x}{(x+6)(x-6) x}-\frac{x-c}{(x+6)(x-6) x} \\ & =\frac{d x-x+e}{(x+6)(x-6) x} \\ & =\frac{x+f}{(x+6)(x-6) x} \\ & =\frac{g}{x(x-6)}\end{aligned} \)

We start with the expression \[ \frac{2}{x^2-36}-\frac{1}{x^2+6x}. \] **Step 1. Factor the denominators.** \[ x^2-36=(x+6)(x-6) \] \[ x^2+6x=x(x+6). \] Thus, the expression becomes \[ \frac{2}{(x+6)(x-6)}-\frac{1}{x(x+6)}. \] In the given setup, the second denominator is written as \(x(x+a)\). To match, we have \[ x+a=x+6 \quad\Longrightarrow\quad a=6. \] **Step 2. Write both fractions with the common denominator \(x(x+6)(x-6)\).** Multiply the numerator and denominator of the first fraction by \(x\): \[ \frac{2}{(x+6)(x-6)}=\frac{2x}{x(x+6)(x-6)}. \] For the second fraction, multiply numerator and denominator by \(x-6\): \[ \frac{1}{x(x+6)}=\frac{x-6}{x(x+6)(x-6)}. \] Now the expression is \[ \frac{2x}{x(x+6)(x-6)}-\frac{x-6}{x(x+6)(x-6)}. \] This is written in the problem as \[ \frac{2x}{(x+6)(x-6)x}-\frac{x-c}{(x+6)(x-6)x}. \] Comparing the second numerator, we have \[ x-c=x-6 \quad\Longrightarrow\quad c=6, \] and the first numerator gives \[ 2x \quad\Longrightarrow\quad b=2. \] **Step 3. Combine the fractions.** Subtracting the numerators \[ 2x-(x-6)=2x-x+6=x+6. \] This combined fraction is written as \[ \frac{x+6}{(x+6)(x-6)x}. \] According to the given sequence, the combined numerator is also written as \[ dx - x + e. \] Setting \[ dx - x + e = x+6, \] rewrite \(dx - x\) as \((d-1)x\). Therefore, \[ (d-1)x + e = x+6. \] This equality holds for all \(x\) if \[ d-1=1 \quad\Longrightarrow\quad d=2, \] \[ e=6. \] **Step 4. Express in the form \(\frac{x+f}{(x+6)(x-6)x}\).** We now equate \[ x+f=x+6 \quad\Longrightarrow\quad f=6. \] **Step 5. Write the final simplified expression.** The common factor \((x+6)\) in the numerator and denominator cancels (assuming \(x\neq -6\)), so \[ \frac{x+6}{(x+6)(x-6)x}=\frac{1}{x(x-6)}. \] This is written in the problem as \[ \frac{g}{x(x-6)} \quad\Longrightarrow\quad g=1. \] **Final Answer:** \[ a=6,\quad b=2,\quad c=6,\quad d=2,\quad e=6,\quad f=6,\quad g=1. \]