I can't wrap my head around this. Let's say I have this example
$256 + 765x + 864x^2 + 432x^3 + 81x^4 = 0$
Now I've already checked the solution which is $-\frac{4}{3}$. However I can't figure out what to factor where so that I get closer to a solution. During the solving steps you even have to write out a number as a sum of numbers to be able to factor it out. For example I should do
$256 + 765x + 864x^2 + 324x^3 + 108x^3 + 81x^4 = 0$
Where I split the $432x^3$ so that I could factor out $27x^3$ from $108x^3 + 81^4$ which would give me
$256 + 765x + 8624x^2 + 324x^3 + 27x^3(4+3x) = 0$
And factor out $(4+3x)$ in a similar manner from the rest. Now I can't look at the example and think "oh I can factor out $27x^3$ here easy peasy. Clearly I am not good enough with this type of problems. Would you be so kind and give me some general tips and how should I practice these?
Thank you
To summarise the comments, the most plausible answer is that the polynomial is indeed
$$ f=(3x+4)^4=81x^4 + 423x^3 + 864x^2 + 768x + 256. $$
By the rational root theorem, one finds the linear factor $3x+4$. Then, by long division, one obtains the quotient
$$ 27x^3 + 108x^2 + 144x + 64. $$
Again by the rational root theorem, there is a linear factor $3x+4$ with quotient $$ 9x^2+24x+16. $$ And again once more!