Simplify: \( \frac{x^{2}+4 x}{x^{2}+12 x+32}-\frac{9 x+36}{x^{2}+12 x+32} \)

To simplify the expression \[ \frac{x^{2}+4 x}{x^{2}+12 x+32}-\frac{9 x+36}{x^{2}+12 x+32}, \] we start by noting that both terms share a common denominator, \( x^{2} + 12 x + 32 \). We can combine the numerators over this common denominator: \[ \frac{(x^{2}+4 x) - (9 x + 36)}{x^{2}+12 x + 32}. \] Next, let's simplify the numerator: \[ x^{2} + 4x - 9x - 36 = x^{2} - 5x - 36. \] Now our expression looks like this: \[ \frac{x^{2} - 5x - 36}{x^{2} + 12x + 32}. \] The next step is to factor both the numerator and the denominator, if possible. Starting with the numerator \(x^{2} - 5x - 36\), we look for two numbers that multiply to \(-36\) and add to \(-5\). These numbers are \(-9\) and \(4\), allowing us to factor the numerator as: \[ (x - 9)(x + 4). \] Now, for the denominator \(x^{2} + 12x + 32\), we look for two numbers that multiply to \(32\) and add to \(12\). These numbers are \(8\) and \(4\), allowing us to factor the denominator as: \[ (x + 8)(x + 4). \] Now, we rewrite the entire expression as: \[ \frac{(x - 9)(x + 4)}{(x + 8)(x + 4)}. \] We see that \(x + 4\) cancels out (as long as \(x \neq -4\)): \[ \frac{x - 9}{x + 8}. \] So, the final simplified expression is: \[ \frac{x - 9}{x + 8}, \quad x \neq -4. \]