Solve the equation and check the solution.

$4|3x+4|=4x+8$

For any real numbers $a,b$, where $b\ge 0$, if $\left|a\right|=b$ then $a=b$ or $a=-b$.

Consider the equation $4|3x+4|=4x+8$.

Divide both sides of the equation by 4.

Use the concept of absolute value equation as follows:

$3x+4=-\left(x+2\right)$or $3x+4=x+2$.

Case 1. Simplify $3x+4=-\left(x+2\right)$.

Simplify the equation as follows:

$\begin{array}{l}3x+4=-\left(x+2\right)\\ 3x+4=-x-2\\ 3x+x=-2-4\\ 4x=-6\\ x=-\frac{6}{4}\\ x=-\frac{3}{2}\end{array}$

Case 2. Simplify $3x+4=x+2$.

Simplify the equation as follows:

$\begin{array}{l}3x+4=x+2\\ 3x-x=2-4\\ 2x=-2\\ x=-1\end{array}$

Therefore, the solution is $x=-1,-\frac{3}{2}$.

Case 1. Substitute $x=-1$in the equation $4|3x+4|=4x+8$and simplify as follows:

$\begin{array}{l}4|3x+4|=4x+8\\ 4|3\left(-1\right)+4|=4\left(-1\right)+8\\ 4|-3+4|=-4+8\\ 4\left|1\right|=4\\ 4=4\end{array}$

Case 2. Substitute $x=-\frac{3}{2}$in the equation $4|3x+4|=4x+8$and simplify as follows:

$\begin{array}{l}4|3x+4|=4x+8\\ 4|3\left(-\frac{3}{2}\right)+4|=4\left(-\frac{3}{2}\right)+8\\ 4|-\frac{9}{2}+4|=-6+8\\ 4\left|\frac{-9+8}{2}\right|=2\\ 4\left|-\frac{1}{2}\right|=2\\ 4\left(\frac{1}{2}\right)=2\\ 2=2\end{array}$

Since the left-hand side of the equation is equal to the right-hand side of the equation in both the cases therefore, the solution is $x=-1,-\frac{3}{2}$.