Let for $a\neq { a }_{ 1 }\neq 0$
$f(x)={ ax }^{ 2 }+bx+c,g(x)={ a }_{ 1 }{ x }^{ 2 }+{ b }_{ 1 }x+{ c }_{ 1 }$ and $p(x)=0$ only for x = -1 and p(-2) = 2, the the value of p(2) is :

$\begin{array}{l}a \neq a_{1} \\ P(-2)=2\end{array}$

$\begin{aligned} P(x) & =\left(a x^{2}+b x+c\right)-\left(a_{1} x^{2}+b_{1} x+c_1\right) \\ & =\left(a-a_{1}\right) x^{2}+\left(b-b_{1}\right) x+\left(c-c_{1}\right)\end{aligned}$

$\begin{array}{l}P(x)=0, x=-1 \\ \text { Sum of roots }=-2 \\ \frac{-\left(b-b_{1}\right)}{a-a_{1}}=-2 \\ \Rightarrow b-b_{1}=2\left(a-a_{1}\right)\end{array}$

$\begin{array}{l}\text { Product of roots }=1 \\ \frac{a-c_{1}}{a-a_{1}}=1 \\ c-c_{1}=a-a_{1} \\\end{array}$

$\begin{aligned} P(-2) & =\left(a-a_{1}\right) \cdot(-2)^{2}+\left(b-b_{1}\right)(-2)+c-c_{1} \\ 2 & =4\left(a-a_{1}\right)-2\left(b-b_{1}\right)+\left(c-c_{1}\right) \\ 2 & =a-a_{1}\end{aligned}$

$\begin{aligned} P(2) & =4\left(a-a_{1}\right)+2\left(2\left(a-a_{1}\right)+\left(a-a_{1}\right)\right. \\ & =9 \times 2=18 .\end{aligned}$