For the function: $f(x)=2 x^{2}+3 x-9$, find and simplify a. $\frac{f(x)-f(a)}{x-a}=$ b. $\frac{f(x+h)-f(x)}{h}=$

Step 1 ：The function given is \(f(x)=2 x^{2}+3 x-9\).

Step 2 ：For part a, we need to find the difference quotient \(\frac{f(x)-f(a)}{x-a}\).

Step 3 ：Substitute \(x\) and \(a\) into the function to get \(f(x) = 2x^{2} + 3x - 9\) and \(f(a) = 2a^{2} + 3a - 9\).

Step 4 ：Subtract \(f(a)\) from \(f(x)\) to get \(f(x) - f(a) = 2x^{2} - 2a^{2} + 3x - 3a\).

Step 5 ：Divide the result by \(x - a\) to get the difference quotient for part a: \(\frac{f(x)-f(a)}{x-a} = 2a + 2x + 3\).

Step 6 ：For part b, we need to find the difference quotient \(\frac{f(x+h)-f(x)}{h}\).

Step 7 ：Substitute \(x+h\) and \(x\) into the function to get \(f(x+h) = 2(x+h)^{2} + 3(x+h) - 9\) and \(f(x) = 2x^{2} + 3x - 9\).

Step 8 ：Subtract \(f(x)\) from \(f(x+h)\) to get \(f(x+h) - f(x) = 2h^{2} + 4hx + 3h\).

Step 9 ：Divide the result by \(h\) to get the difference quotient for part b: \(\frac{f(x+h)-f(x)}{h} = 2h + 4x + 3\).

Step 10 ：So, the final answers are: \(\frac{f(x)-f(a)}{x-a} = \boxed{2a + 2x + 3}\) and \(\frac{f(x+h)-f(x)}{h} = \boxed{2h + 4x + 3}\).