Question number 10.

Use synthetic division to evaluate $P(-4)$ for
\[
P(x)=x^{3}-4 x^{2}-9 x+36
\]
18
21
26
10
16

Step 1 ：Write down the coefficients of the polynomial \(P(x)\), which are 1, -4, -9, and 36.

Step 2 ：Write down the value we are evaluating the polynomial at, which is -4.

Step 3 ：Set up the synthetic division as follows: \[-4 | 1 -4 -9 36\]

Step 4 ：Perform the synthetic division: \[-4 | 1 -4 -9 36\] \[\phantom{-4 |} -4 32 -92\] \[\phantom{-4 |}-----------------\] \[\phantom{-4 |} 1 -8 23 -56\]

Step 5 ：The numbers in the bottom row are the coefficients of the quotient polynomial, and the last number, -56, is the remainder.

Step 6 ：Since we are evaluating \(P(-4)\), the remainder is the value of the polynomial at \(x=-4\).

Step 7 ：\(P(-4) = -56\)

Step 8 ：Therefore, none of the options 18, 21, 26, 10, 16 are correct.

Step 9 ：\(\boxed{P(-4) = -56}\) is the correct answer.