The expression {x}^{3} + 4 {x}^{2} + ax + b has a factor x - 2and leaves a remainder of 35 when divided by x + 3 find the values of a and b

Answers are:     a = -10, b = -4======we use the remainder theorem to solve this question. with the remainder theorem, we set up a system of equations.remainder theorem in a nutshell: the remainder of dividing the polynomial p(x) by x-a is p(a) (plug that a value into the polynomial).if x³ + 4x² + ax + b has a factor of x - 2, this tells us that when we divide  x³ + 4x² + ax + b by x - 2, we get a remainder of 0. so if we plug in 2 into the polynomial for all x, that should be equal to 0 (remainder theorem):     if x³ + 4x² + ax + b leaves a remainder of 35 when we divide it by x+3, then by the remainder theorem, substituting in -3 in for x into the polynomial should equal 35:     we have the following system of equations:     can solve by elimination. multiply both sides of equation (i) by -1    add (i) and (ii) together    we can use either equation to figure out b.we can use (ii):     a = -10, b = -4