To find \((f/g)(x)\), we first need to compute the expression:
\[ \frac{f(x)}{g(x)} = \frac{2x^2 - x - 6}{x^2 + 3x - 10} \]
Next, we will factor both \(f(x)\) and \(g(x)\) to simplify the expression.
Step 1: Factor \(f(x) = 2x^2 - x - 6\)
To factor \(f(x)\), we look for two numbers that multiply to \(2 \cdot (-6) = -12\) and add up to \(-1\). These numbers are \(3\) and \(-4\). Thus, we can rewrite \(f(x)\):
\[ f(x) = 2x^2 - 4x + 3x - 6 = 2x(x - 2) + 3(x - 2) = (2x + 3)(x - 2) \]
Step 2: Factor \(g(x) = x^2 + 3x - 10\)
To factor \(g(x)\), we need two numbers that multiply to \(-10\) and add to \(3\). These numbers are \(5\) and \(-2\). Thus,
\[ g(x) = (x + 5)(x - 2) \]
Step 3: Simplify \((f/g)(x)\)
Now we can rewrite the expression for \((f/g)(x)\):
\[ \frac{f(x)}{g(x)} = \frac{(2x + 3)(x - 2)}{(x + 5)(x - 2)} \]
Notice that \(x - 2\) is a common factor. We can simplify it, but we need to keep in mind that \(x\) cannot be equal to \(2\) because it would make the denominator zero:
\[ (f/g)(x) = \frac{2x + 3}{x + 5}, \quad x \neq 2 \]
Step 4: Find the domain of \((f/g)(x)\)
The domain of \((f/g)(x)\) consists of all real numbers except where the denominator is zero. The denominator of \((f/g)(x)\) is \(g(x) = (x + 5)(x - 2)\).
Setting \(g(x)\) to zero to find the restrictions:
\[ x + 5 = 0 \implies x = -5 \] \[ x - 2 = 0 \implies x = 2 \]
Therefore, the function \((f/g)(x)\) is undefined at \(x = -5\) and \(x = 2\).
Conclusion
The domain of \((f/g)(x)\) is:
\[ \text{Domain: } (-\infty, -5) \cup (-5, 2) \cup (2, \infty) \]
So, we can summarize:
\[ (f/g)(x) = \frac{2x + 3}{x + 5} \quad \text{for } x \neq -5, 2 \]
And the domain is \( (-\infty, -5) \cup (-5, 2) \cup (2, \infty) \).