We are given two functions $f(x) = \sqrt{x+5}$ and $g(x) = (x+2)(x-3)$, and we need to compute the following:

a. $(f+g)(x)$

This is the sum of the two functions: $(f+g)(x) = f(x) + g(x)$ Substitute the values of $f(x)$ and $g(x)$: $(f+g)(x) = \sqrt{x+5} + (x+2)(x-3)$ Expand $g(x)$: $g(x) = (x+2)(x-3) = x^2 - 3x + 2x - 6 = x^2 - x - 6$ Thus, $(f+g)(x) = \sqrt{x+5} + x^2 - x - 6$

b. $(f-g)(x)$

This is the difference of the two functions: $(f-g)(x) = f(x) - g(x)$ Substitute the values of $f(x)$ and $g(x)$: $(f-g)(x) = \sqrt{x+5} - (x^2 - x - 6)$ Simplify: $(f-g)(x) = \sqrt{x+5} - x^2 + x + 6$

c. $(f \cdot g)(x)$

This is the product of the two functions: $(f \cdot g)(x) = f(x) \cdot g(x)$ Substitute the values of $f(x)$ and $g(x)$: $(f \cdot g)(x) = \sqrt{x+5} \cdot (x^2 - x - 6)$

d. $\left( \frac{f}{g} \right)(x)$

This is the quotient of the two functions: $\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)}$ Substitute the values of $f(x)$ and $g(x)$: $\left( \frac{f}{g} \right)(x) = \frac{\sqrt{x+5}}{x^2 - x - 6}$

Would you like me to provide further details or evaluate any specific part?

Here are 5 related questions to consider:

1. What is the domain of the function $f(x) = \sqrt{x+5}$?
2. How does expanding $g(x) = (x+2)(x-3)$ affect the combined functions?
3. What is the significance of function composition in algebraic functions?
4. How can we find the points of intersection between $f(x)$ and $g(x)$?
5. What methods are used to simplify or solve the quotient $\frac{f(x)}{g(x)}$?

Tip: Always check the domain restrictions for combined functions, especially when dealing with square roots and denominators.