(a) Polynomial division gives \[\begin{array}{r|l}\text{quotient term}&\text{remaining polynomial}\\\hline x^3&4x^2-8x\\2x&-14x\\-7&21\end{array}\] Hence \[p(x)=(2x+3)(x^3+2x-7)+21.\] The quotient is \(\boxed{x^3+2x-7}\) and the remainder is \(\boxed{21}\).

(b) \(p(x)-21=(2x+3)(x^3+2x-7)\). The negative crossing is \(\boxed{\alpha=-3/2}\). The cubic has positive derivative \(3x^2+2\), so its unique root is the positive crossing \(\beta\). It satisfies \[x^3+2x-7=0\quad\Rightarrow\quad\boxed{x=\sqrt[3]{7-2x}}.\]

(c) For \(f(x)=x^3+2x-7\), \[f(1.5)=-0.625<0,\qquad f(1.6)=0.296>0.\] Continuity and this sign change show that \(\boxed{1.5<\beta<1.6}\).

(d) Use \(x_{n+1}=\sqrt[3]{7-2x_n}\), starting with \(x_0=1.5\). Keep full precision internally and display 6 significant figures:

\[\begin{array}{c|rrrrrrrrr}n&1&2&3&4&5&6&7&8&9\\\hline x_n&1.58740&1.56393&1.57030&1.56858&1.56905&1.56892&1.56895&1.56894&1.56895\end{array}\]

The iterations settle to the same 4-significant-figure value: \(\boxed{\beta=1.569}\).