What is the time complexity of an algorithm that calls a polynomial time algorithm?
I assume $n$ is the number of entries in array/set A.


Yes, since polynomials are closed against multiplication; specifically, $n(n-1) \cdot p(n)$ is always polynomially bounded as long as $p(n)$ is polynomially bounded.
As a consequence, the only way to achieve exponential running-time by calling a polynomial-time algorithm with parts of the original input is to call it exponentially often.

Example:

Foo(A)
  if |A| &gt;= 2
    a = A[0]
    b = A[1]
    Polynomial-Time-Algorithm-Bar(A, a, b)
    Foo(A - a)
    Foo(A - b)


The number of calls to PTA-Bar is given by

$\qquad\begin{align*}
  C(0) &amp;= 0,\\
  C(1) &amp;= 0,\\
  C(n) &amp;= 2C(n-1) + 1, \qquad n \geq 2,
\end{align*}$

which solves to 
$C(n) = 2^{n-1} - 1$ for $n \geq 1$. Ergo, the running time of Foo is in $\Omega\bigl( 2^n \cdot p(n) \bigr)$ if PTA-Bar takes time $p(n)$.