Consider an unknown polynomial which when divided by (x - 3) and (x - 4) leaves remainders 2 and 1, respectively. Let R(x) be the remainder when this polynomial is divided by (x - 3) (x - 4). If equation R(x)=x2+ax+1has two distinct real roots, then exhaustive values of a areIf R(x)=px2+(q−1)x+6has no distinct real roots and p > 0, then the least value of 3p + q isRange of f(x)=[R(x)]/x2−3x+2 is

Let unknown polynomial be P(x).Let Q(x) and R(x) be the quotient and remainder, respectively, when it is divided by (x - 3X(x - 4). Then, P(x)=(x−3)(x−4)Q(x)+R(x)Then, we have R(x)=ax+b⇒P(x)=(x−3)(x−4)Q(x)+ax+bGiven that P(3) = 2 and P(4) = 1. Hence,3a + b = 2 and 4a + b = 1⇒ a=−1 and b=5⇒ R(x)=5−x5−x=x2+ax+1⇒x2+(a+1)x−4=0Given that roots are real and distinct. Therefore,D>0⇒(a+1)2+16>0which is true for all real a.−x+5=px2+(q−1)x+6⇒px2+qx+1=0Now, p > 0 and equation has no distinct real roots or equation has real and equal or imaginary roots. Then,px2+qx+1≥0,∀x∈R⇒ f(3)≥0⇒9p+3q+1≥0⇒3p+q≥−1/3Hence, the least value of 3p+q is −1/3.f(x)=y=−x+5x2−3x+2⇒ yx2+(1−3y)x+2y−5=0Now, x is real, then      D≥0⇒ (1−3y)2  −4y(2y−5)≥0or    y2+14y+1≥0or    y∈−∞,−14−1922∪−14+1922,∞    (−∞,−7−43]∪[−7+43,∞)