Answer: The remainder is \(-24\).

Answer: The remainder is \(-24\).

Since \(x-3\) is a factor, the factor theorem gives

\(p(3)=0\).

So

\(27m-17(9)+3n+6=0\),

which simplifies to

\(27m+3n=147\),

or

\(9m+n=49\). \(\quad (1)\)

When \(p(x)\) is divided by \(x+1\), the remainder is \(p(-1)\). Hence

\(p(-1)=-12\).

Therefore

\(-m-17-n+6=-12\),

so

\(m+n=1\). \(\quad (2)\)

Subtracting (2) from (1) gives

\(8m=48\),

so \(m=6\). Then \(n=1-6=-5\).

The remainder when \(p(x)\) is divided by \(x-2\) is \(p(2)\):

\(p(2)=6(8)-17(4)-5(2)+6\).

Thus

\(p(2)=48-68-10+6=-24\).