Select the simplified form of this expression. \[ \frac{6 x+21}{x^{2}+5 x+4}-\frac{5}{x+1} \]

To simplify the given expression \[ \frac{6 x+21}{x^{2}+5 x+4}-\frac{5}{x+1}, \] let's start with the first fraction. The denominator \(x^2 + 5x + 4\) can be factored as \((x+1)(x+4)\). Thus, we can rewrite the expression as: \[ \frac{6x + 21}{(x+1)(x+4)} - \frac{5}{x+1}. \] Next, to combine these two fractions, we need a common denominator which is \((x+1)(x+4)\). We can write the second fraction with this common denominator: \[ \frac{5}{x+1} = \frac{5(x+4)}{(x+1)(x+4)}. \] Now the expression becomes: \[ \frac{6x + 21}{(x+1)(x+4)} - \frac{5(x+4)}{(x+1)(x+4)}. \] We can combine the fractions: \[ \frac{(6x + 21) - 5(x + 4)}{(x+1)(x+4)}. \] Expanding the numerator: \[ (6x + 21) - 5(x + 4) = 6x + 21 - 5x - 20 = (6x - 5x) + (21 - 20) = x + 1. \] So now we have: \[ \frac{x + 1}{(x+1)(x+4)}. \] Next, we simplify the fraction by cancelling out the \(x + 1\) term (assuming \(x \neq -1\)): \[ \frac{1}{x + 4}. \] Thus, the simplified form of the expression is: \[ \frac{1}{x + 4}, \quad (x \neq -1). \]