# If the expression mx^4 + nx^3 - xÂ² + 2x + 3 has remainder 4x + 3 when divided byX^2+X-2, find the value m and n.?

$As\;x^2+x-2=(x+2)(x-1)\\*Let\;f(x)=mx^4+nx^3-x^2+2x+3=(x^2+x2) Q(x) +4x+3,\\*where\;Q(x)\;is\;the\;quotient \\* Then,\\* f(-2)=m(-2)^4+n(-2)^3-(-2)^2+2(-2)+3=4(-2)+3\\*\Rightarrow 16m-8n-4-4+3=-5 \\*\Rightarrow 2m-n=0 ......... (1) \\*And,\; f(1)=m+n-1+2+3=4(1)+3\\* \Rightarrow m+n=3 ........ (2)\\* Solving \;(1)\;and\;(2)\;simultaneously,\;we\;get\\* m=1,n=2\\*$

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