Let $f(x)=2 x^{2}-7 x+5$ and $g(x)=x^{2}+16$. Find $f+g_{1} f-g_{1} f \cdot g$, and $\frac{f}{g}$. Simplify your answers.
1. $f+g=3 x^{2}-7 x+21$
2. $f-g=$
3. $f \cdot g=$
4. $\frac{f}{g}=$

Step 1 ：First, we need to find the sum of the functions $f$ and $g$. This is done by adding the corresponding terms of the two functions together.

Step 2 ：$f+g=(2x^{2}-7x+5)+(x^{2}+16)=3x^{2}-7x+21$

Step 3 ：Next, we need to find the difference of the functions $f$ and $g$. This is done by subtracting the corresponding terms of $g$ from $f$.

Step 4 ：$f-g=(2x^{2}-7x+5)-(x^{2}+16)=x^{2}-7x-11$

Step 5 ：Then, we need to find the product of the functions $f$ and $g$. This is done by multiplying the corresponding terms of the two functions together.

Step 6 ：$f \cdot g=(2x^{2}-7x+5) \cdot (x^{2}+16)=2x^{4}-7x^{3}+5x^{2}+32x^{2}-112x+80$

Step 7 ：Finally, we need to find the quotient of the functions $f$ and $g$. This is done by dividing the corresponding terms of $f$ by $g$.

Step 8 ：$\frac{f}{g}=\frac{2x^{2}-7x+5}{x^{2}+16}$

Step 9 ：This cannot be simplified further, so the final answers are $f+g=3x^{2}-7x+21$, $f-g=x^{2}-7x-11$, $f \cdot g=2x^{4}-7x^{3}+37x^{2}-112x+80$, and $\frac{f}{g}=\frac{2x^{2}-7x+5}{x^{2}+16}$