I cant seem to find a way to prove it because:
if Ax = b has a solution then we know that if we multiply on left side by matrix A we will get A(Ax) = Ab witch then we can replace Ax by b so Ab = Ab holds true.
but i am not sure this means the linear equations (A^2)x = b has a solution.
You cannot prove it as the statement is not true for all instances of $A$ and $b$. For example, let $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$, and let $b= \begin{bmatrix}1 \\ 0 \end{bmatrix}$. Then $Ax=b$ has a solution. But what about the equation $A^2x=b$? In fact, can you note that the matrix $A^2$ is $0$.