Use synthetic division to divide $f(x)$ by $x-c$ then write $f(x)$ in the form $f(x)=(x-c) q(x)+r$. \[ f(x)=x^{3}+9 x^{2}+10 x+5 ; \quad x-2 \]

Step 1 ：Write down the coefficients of the polynomial $f(x)=x^{3}+9 x^{2}+10 x+5$, which are 1, 9, 10, and 5.

Step 2 ：Perform synthetic division with these coefficients and the value $c=2$.

Step 3 ：The synthetic division process gives us the coefficients of the quotient polynomial and the remainder.

Step 4 ：The coefficients of the quotient polynomial are 1, 11, and 32, and the remainder is 69.

Step 5 ：Use these to write $f(x)$ in the form $f(x)=(x-c) q(x)+r$.

Step 6 ：\(f(x)=(x - 2)(1x^{2} + 11x + 32) + 69\)

Step 7 ：\(\boxed{f(x)=(x - 2)(1x^{2} + 11x + 32) + 69}\)