If x + 2  and x − 1 are the factors of x3 + 10x2 + mx + n, then the values of m and n are respectively

To find the values of m and n, we can use the fact that if x + 2 and x - 1 are factors of x^3 + 10x^2 + mx + n, then when we substitute x = -2 and x = 1 into the expression, the result should be zero.

Step 1: Substitute x = -2 into the expression: (-2)^3 + 10(-2)^2 + m(-2) + n = 0 -8 + 40 + (-2m) + n = 0 32 - 2m + n = 0

Step 2: Substitute x = 1 into the expression: (1)^3 + 10(1)^2 + m(1) + n = 0 1 + 10 + m + n = 0 11 + m + n = 0

Step 3: Solve the system of equations formed by the two equations from steps 1 and 2. Subtract the second equation from the first equation to eliminate n: (32 - 2m + n) - (11 + m + n) = 0 - 0 32 - 2m + n - 11 - m - n = 0 21 - 3m = 0 -3m = -21 m = 7

Step 4: Substitute the value of m into one of the original equations to solve for n. Let's use the second equation from step 2: 11 + m + n = 0 11 + 7 + n = 0 18 + n = 0 n = -18

Therefore, the values of m and n are 7 and -18, respectively.