Find all numbers that must be excluded from the domain of the rational expression. (x-2)/(x^2)+5x+4

To find the numbers that must be excluded from the domain of the rational expression \(\frac{x-2}{x^2+5x+4}\), we need to determine when the denominator is equal to zero, as division by zero is undefined.<br /><br />The denominator of the expression is \(x^2 + 5x + 4\). We set this equal to zero and solve for \(x\):<br /><br />\[<br />x^2 + 5x + 4 = 0<br />\]<br /><br />To solve this quadratic equation, we can factor it. We look for two numbers that multiply to \(4\) (the constant term) and add up to \(5\) (the coefficient of the linear term). These numbers are \(1\) and \(4\).<br /><br />Thus, we can factor the quadratic as:<br /><br />\[<br />x^2 + 5x + 4 = (x + 1)(x + 4)<br />\]<br /><br />Setting each factor equal to zero gives us the solutions:<br /><br />1. \(x + 1 = 0 \Rightarrow x = -1\)<br />2. \(x + 4 = 0 \Rightarrow x = -4\)<br /><br />Therefore, the numbers that must be excluded from the domain of the rational expression are \(x = -1\) and \(x = -4\). These values make the denominator zero, resulting in an undefined expression. <br /><br />In conclusion, the domain of the rational expression excludes \(x = -1\) and \(x = -4\).