If the expression `a x^4+b x^3-x^2+2x+3` has remainder `4x+3` when divided by `x^2+x-2,` find the value of `aa n dbdot`

Explanation

According to the question,
`ax^(4) + bx^(3) - x^(2)+ 2x + 3`
`=(x^(2)+X -2)Q(x)+(4x +3),` where Q(x) is quotient
`=(x-1)(x+2)Q(x)+(4x +3)`
Putting x = 1, we get
a + b - 1 + 2 + 3 = 4(1) + 3
or a + b = 3
Putting x = - 2, we get
` a(-2)^(4) + b(-2)^(3)-(-2)^(2) + 2(-2) + 3 = 4 (-2) + 3`
or 2a - b = 0
Solving (1) and (2), we get a = 1 and b = 2 .