Identify the domain of the expression. (x+5)/(2x^2)+13x+15 The domain is all real numbers except 5 and (3)/(2) The domain is all real numbers except -(3)/(2) The domain is all real numbers except -5 The domain is all real numbers except -5 and -(3)/(2)

## Step 1: Identify the Denominator<br />### The expression is $\frac{x+5}{2x^2 + 13x + 15}$. To find the domain, we need to determine where the denominator is zero, as division by zero is undefined.<br />## Step 2: Solve for Zero in the Denominator<br />### Set the denominator equal to zero: $2x^2 + 13x + 15 = 0$. Solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 2$, $b = 13$, and $c = 15$.<br />## Step 3: Calculate the Discriminant<br />### Calculate the discriminant: $b^2 - 4ac = 13^2 - 4 \cdot 2 \cdot 15 = 169 - 120 = 49$.<br />## Step 4: Find the Roots<br />### Substitute into the quadratic formula: $x = \frac{-13 \pm \sqrt{49}}{4} = \frac{-13 \pm 7}{4}$. This gives two solutions: $x = \frac{-13 + 7}{4} = -\frac{3}{2}$ and $x = \frac{-13 - 7}{4} = -5$.<br />## Step 5: Determine the Domain<br />### The domain of the expression is all real numbers except where the denominator is zero. Therefore, the domain is all real numbers except $x = -\frac{3}{2}$ and $x = -5$.