Year 10 (A1) Mathematics Polynomial Function (II) (b) The remainder when $x^{4}+a x+b$ is divided by $(x-2)(x+1)$ is $x+2$. Find the values of and $b$.

Step 1 ：Find the remainders when the polynomial is divided by (x-2) and (x+1) using the Remainder Theorem: \(f(2) = 2^4 + 2a + b\) and \(f(-1) = (-1)^4 - a + b\)

Step 2 ：Equate the remainders to the given remainder x+2: \(2^4 + 2a + b = 4\) and \((-1)^4 - a + b = 1\)

Step 3 ：Solve the system of linear equations to find the values of a and b: \(a = -4\) and \(b = -4\)

Step 4 ：\boxed{a = -4, b = -4}