I am trying to converge to the root of this equation
$$2 x^3 + 4 x^2 – 2 x – 5 = 0$$
through 3 different arrangements so far tried 4 and only one worked which is this one
$$(x/2 + 5/4 - 0.5 x^3)^{1/2}$$ some of the arrangements I tried got me to the root after too many iterations (5000~15000) so I ignored them How to find more arrangements to try?
Newton's Method gives the iteration $$ \begin{align} x_{k+1} &=x_k-\frac{2x_k^3+4x_k^2–2x_k–5}{6x_k^2+8x_k-2}\tag1\\ &=\frac{4x_k^3+4x_k^2+5}{6x_k^2+8x_k-2}\tag2 \end{align} $$ Iteration $(2)$ could be thought of as a rearrangement since cross multiplication and subtraction gives $$ \begin{align} x&=\frac{4x_k^3+4x_k^2+5}{6x_k^2+8x_k-2}\tag3\\ 0&=2x^3+4x^2-2x-5\tag4 \end{align} $$
Computing the Roots
Using the iteration in $(2)$ and starting at $x_1\in\left\{1,-1,-2\right\}$ gives $$ \begin{array}{r|l} k&x_k\\\hline 1&1\\ 2&1.0833333333333333333\\ 3&1.0781830462681526511\\ 4&1.0781625876515298067\\ 5&1.0781625873293308468\\ 6&\color{#C00}{1.0781625873293308468} \end{array} $$ $$ \begin{array}{r|l} k&x_k\\\hline 1&-1\\ 2&-1.2500000000000000000\\ 3&-1.3095238095238095238\\ 4&-1.3153867041891925251\\ 5&-1.3154487998979105398\\ 6&-1.3154488069075722138\\ 7&-1.3154488069075723031\\ 8&\color{#C00}{-1.3154488069075723031} \end{array} $$ $$ \begin{array}{r|l} k&x_k\\\hline 1&-2\\ 2&-1.8333333333333333333\\ 3&-1.7724867724867724868\\ 4&-1.7629503938578455289\\ 5&-1.7627139251465655298\\ 6&-1.7627137804218127461\\ 7&-1.7627137804217585437\\ 8&\color{#C00}{-1.7627137804217585437} \end{array} $$