Question

Let $f\left(x\right)$ and $g\left(x\right)$ be two quadratic polynomials with real coefficients such that their leading coefficients are always different. $h\left(x\right)$ is another polynomial which satisfies $h\left(x\right)=f\left(x\right)-g\left(x\right)\forall x\in R.$ If $h\left(x\right)=0$ only at $x=-3$ and $h(-1)=6,$ then the value of $h\left(5\right)$ is

Answer 1

Let $f\left(x\right)=a_1{x}^2+b_{1}x+c_1$
and $g\left(x\right)=a_2{x}^2+b_{2}x+c_2$
where $a_1\ne a_2\ne 0$
$h\left(x\right)$ is also a quadratic polynomial as $a_1\ne a_2$
Since, $h\left(x\right)=0$ only at $x=-3$,
$x=-3$ is a repeated root of $h\left(x\right)=0$.

Let $h\left(x\right)=k(x+3{)}^2,$ where $k$ is a non-zero constant.
$h(-1)=6\Rightarrow k(-1+3{)}^2=6$
$\Rightarrow k=\frac{3}{2}$
$\Rightarrow h\left(x\right)=\frac{3}{2}(x+3{)}^2$
$\therefore h\left(5\right)=\frac{3}{2}(5+3{)}^2=96$