$f(x)$ = $ax^2+bx+c$
$f(2)$ = $4a+2b+c$
$2$ = $4a+2b+c$.....$(1)$
$f(1)$ = $a+b+c$
$3$ = $a+b+c$.....$(2)$
$f^\prime$$(x)$ = $2ab+b$
$f^\prime$$(1)$ = $2a+b$
$0$ = $2a+b$.....$(3)$
$(1)$-$(2)$; -$1$ = $3a+b$
$(4)$-$(3)$; $a$ = -$1$
$(3)$;  $0$ = $2(-1)+b$
            $b$ = $2$
$(2)$; $3$ = $-1+2+c$
           $3$ = $1+c$
           $c$ = $2$
$\int\limits_{-1}^{2}$$f(x)$$dx$ = $\int\limits_{-1}^{2}$$-x^2+2x+2$$dx$
= $\frac{-8}{3}$+$4+4$-$\frac{1}{3}$+$1-2$
= $\frac{16}{3}$-$\frac{-2}{3}$
= $\frac{18}{3}$ = $6$