$x^2+x+2\equiv 0 \pmod {56}$ using CRT

Question

If we have for example the equation: $x^2+x+2 \equiv 0 \pmod{56}$, I understand that we need to divide to 2 equations:

1. $x^2+x+2\equiv 0\pmod{8}$
2. $x^2+x+2 \equiv 0 \pmod{7}$

Then, how we unite the two results for the original equation?

Answer 1

Since you've already mentioned that you have find that $$x\equiv 2,5 \text{ mod } 8 \text{ and } x\equiv 3 \text{ mod } 7 $$ are the solution then you've to find the solution that satisfy both.
Hence for the first congruency if 2 is the solution then so is $2+8n$ hence 10 also satisfy the same.
Now look for the second congruency , 3 is the solution then so is $3+7n$ hence 10 also satisfy the same. Therefore , $$x\equiv 10 \text{ mod } 56$$ Now again go back to first congruency. As 5 is the solution then so is $5+8n$ therefore try plug in the values so that this number is 3 more than a multiple of 7. Plug in $n=5$ gives $45$ which satisfy the first congruency. But also $45=7\cdot 6+3$ hence this is the other solution.
Therefore the solution are : $$x\equiv 10,45\text{ mod } 56$$