Solve $2x^4 + 7x^3 -34x^2 -21x + 18 = 0$ over the real numbers.
I know the final answers but I want a logical way to solve it. I must add you can't use derivation or other advanced formulas. You can use ways such multiplying two sides or using quadratic formulas and such ways. Also you can use simple factorization methods or changing the variable but no advanced calculus or differential formulas.
I want to know a clear way to solve it. For example, if you multiply it by a number or variable, say the reason for doing that and how did you find the appropriate number (or polynomial).
Thanks and sorry for my English.
For higher degree polynomials, I recommend first checking with the rational roots theorem.
It's very simple, take the first coefficient $(2)$ and the last coefficient $(18)$, and factor them:
$$2=1\times2$$
$$18=1\times2\times3^2$$
Thus, all real rational roots are of the form
$$r=\pm\frac{\{1,2,3,6,9,18\}}{\{1,2\}}$$
So, we run through and test:
$$r\ne+1\\\boxed{r=-1}\\\boxed{r=+\frac12}\\r\ne-\frac12\\r\ne+2\\r\ne-2\\\boxed{r=+3}\\r\ne-3\\r\ne+\frac32\\r\ne-\frac32\\r\ne+6\\\boxed{r=-6}$$
And look at that, we've got all four roots! So we can then put it back into factored form:
$$2x^4+7x^3-34x^2-21x+18=2(r+1)(r-\frac12)(r-3)(r+6)$$