All $a$ that equation has at least one root. $a^2+7|x+1|+5 \sqrt{x^2+2x+5}=2a+3|x-4a+1|$

Question

Find all $a$ such that the equation has at least one root.
$$a^2+7|x+1|+5 \sqrt{x^2+2x+5}=2a+3|x-4a+1|$$ What have I done:
substitution $t=x+1$ and some rearrangements $$(a-1)^2-1+7|t|+5\sqrt{t^2+4}-3|t-4a|=0$$ I solved a lot of equations like this, but now I can't easily see the way through.
Usually I noticed something like that the square root $\sqrt{m^2+36} \ge 6$. Then usually opened up the modulus considering two different cases $t>4a$ and other. Then found that $7|t|+3t \ge 0$ and $7|t|-3t \ge 0$ . It helped me along with rearrangements that made left side bigger or equal to one number, the right less or equal to the same number...
But now nothing helps.
Please show your thinking process as thorough as you can. Show how you came up to that way of solution.

Answer 1

$3|t-4a|-7|t|$ is at its largest at $t=0$.
$(a-1)^2-1+5\sqrt{t^2+4}$ is at its least when $t=0$.
So your expression it at its least when $t=0$.
There is a solution if and only if the value at $t=0$ is negative or zero.