Answer: remainder \(24\); \(p(x)=(x+2)(3x-1)(5x-1)\).

(a) By the remainder theorem, the remainder when \(p(x)\) is divided by \(x+1\) is \(p(-1)\).

\(p(-1)=15(-1)^3+22(-1)^2-15(-1)+2.\)

So

\(p(-1)=-15+22+15+2=24.\)

The remainder is therefore \(24\).

(b)(i) To show that \(x+2\) is a factor, evaluate \(p(-2)\):

\(p(-2)=15(-8)+22(4)-15(-2)+2.\)

Thus

\(p(-2)=-120+88+30+2=0.\)

Since \(p(-2)=0\), \(x+2\) is a factor of \(p(x)\).

(b)(ii) Divide \(p(x)\) by \(x+2\):

\(p(x)=(x+2)(15x^2-8x+1).\)

Now factorise the quadratic:

\(15x^2-8x+1=(3x-1)(5x-1).\)

Therefore

\(p(x)=(x+2)(3x-1)(5x-1).\)