I am new to analytical geometry and excuse me for my notations. We have four lines:
$l_1: u_1x + v_1y + r_1 = 0$
$l_2: u_2x + v_2y + r_2 = 0$
$m_1: u_3x + v_3y + r_3 = 0$
$m_2: u_4x + v_4y + r_4 = 0$
We know that $l_1$ and $l_2$ cross int $(l_1, l_2)$ and $m_1$ and $m_2$ cross in $(m_1, m_2)$
Find the straight line $p$ which is equal to $p = (l_1, l_2)(m_1, m_2)$.
from $(l_1, l_2 ) \Rightarrow u_1x + v_1y + r_1 + u_2x + v_2y + r-2 = 0$
from $(r_1, r_2 ) \Rightarrow u_3x + v_3y + r_3 + u_4x + v_4y + r_4 = 0$
right?
if so I don't understand what the notation in $p = (l_1, l_2)(m_1, m_2 )$, really means, actually I don't quite undestand the idea that $(l_1, l_2 ) \Rightarrow l_1 + l_2 = 0$
If $(l_1, l_2)$ is for the intersection $I$ of $l_1$ and $l_2$;
If $(m_1, m_2)$ is for the intersection $J$ of $m_1$ and $m_2$;
Notation $p = (l_1, l_2)(m_1, m_2 )$ which is is not classical, should mean the straight line $IJ$...
I dont follow you when you write $l_1+l_2=0$. (The "right ?" appeals answer "no!")
What you have to do is
1) solve the linear system of 2 equations formed by $l_1$ and $l_2$ to obtain the coordinates of $I$, then
2) do the same for $J$, then
3) compute the equation of straight line $IJ$.