  Let $A,B,C,D$ be rational numbers such that $ABC \neq 0$, and let
$n_1>n_2>n_3>0$ be positive integers. We solve the equation
  $$ Ax^{n_1}+Bx^{n_2}+Cx^{n_3}+D = f(g(x)),$$ in $f,g \in \mathbb{Q}[x]$. In
sequel we use Bilu-Tichy method to prove finitness of integral solutions of the
equations $$ Ax^{n_1}+Bx^{n_2}+Cx^{n_3}+D = Ey^{m_1}+Fy^{m_2}+Gy^{m_3}+H, $$
where $A,B,C,D,E,F,G,H$ are rational numbers $ABCEFG \neq 0$ and
$n_1>n_2>n_3>0$, $m_1>m_2>m_3>0$, $\gcd(n_1,n_2,n_3) = \gcd(m_1,m_2,m_3)=1$ and
$n_1,m_1 \geq 9$. And the equation $$ A_1x^{n_1}+A_2x^{n_2}+\ldots+A_l x^{n_l}
+ A_{l+1} = Ey^{m_1}+Fy^{m_2}+Gy^{m_3}, $$ where $l \geq 4$ is fixed integer,
$A_1,\ldots,A_{l+1},E,F,G$ are non-zero rational numbers, except for possibly
$A_{l+1}$, $n_1>n_2>\ldots > n_l>0$, $m_1>m_2>m_3>0$ are positive integers such
that $\gcd(n_1,n_2, \ldots n_l) = \gcd(m_1,m_2,m_3)=1$, and $n_1 \geq 4$, $m_1
\geq 2l(l-1)$.
