(Note: I am assuming that there is a minus in front of the 2x because the problem is much more difficult if there is a plus.)
To find solutions to this equation we will need to factor it. The only way to factor this, as far as I can see, is to use trial and error of the possible rational roots.
The possible rational roots of a polynomial are all the possible fractions, positive and negative, that can be formed using a factor of the constant term (at the end) over a factor of the leading coefficient (in front of the term with the highest exponent). Your constant term is -10 and you leading coefficient is 1, So the list of possible rational roots are:
10/1, -10/1, 5/1, -5/1, 2/1, -2/1, 1/1 and -1/1
which simplify to:
10, -10, 5, -5, 2, -2, 1 and -1
Synthetic division is often the method used to check these roots. This is a trial and error method but I will not waste time showing the ones that don't work. First we will try -5:
-5 | 1 8 15 -2 -10
---- -5 -15 0 10
---------------------
1 3 0 -2 0
Above we have divided by (x - (-5)) (or (x+5)). The zero in the lower right corner is the remainder of this division. This means (x+5) divides evenly. This means that (x+5) is a factor of . The rest of the bottom row tells us the other factor. The 1 3 0 -2 translates into .
Our equation is now:
We now look for factors of . Its possible rational roots are: 2, -2, 1 and -1. -1 works:
-1 | 1 3 0 -2
---- -1 -2 2
---------------
1 2 -2 0
So (x - (-1)) or (x+1) is a factor of and the other factor (1 2 -2) is . Our equation is now:
The third factor is a quadratic which will not factor easily. But since it is a quadratic we can use the Quadratic Formula to find the values of x that make it zero. So we can proceed to the next step.
The Zero Product Property tells us that this (or any) product can be zero only if one (or more) of the factors is zero. So:
x+5 = 0 or x+1 = 0 or
The first two are simple to solve. We get x = -5 and x = -1 for solutions. The third equation requires use of the Quadratic Formula:
which simplifies as follows:
In long form this is:
or
So the solutions to your equation are:
x = -5 or x = -1 or or