The x-intercepts occur when the function is equal to zero. Graphically, this is when the graph touches the x-axis, hence "x-intercepts".
x^2+4x+3=0
You can solve this in three ways, factoring, "completing the square", or simply using the quadratic equation (which is the result of completing the square). You did not show Mathieu's work, so I'll complete the square, as it is conceptually very important. (because it is the derivation of the quadratic formula and because it is straight forward when factoring would be nearly impossible like it is in most real world problems...you rarely get simple integer factors outside of the classroom :))
x^2+4x+3=0 subtract 3 from both sides
x^2+4x=-3 halve the linear coefficient, square it, add that value to both sides, in this case, (4/2)^2=2^2=4, so add 4 to both sides
x^2+4x+4=1 now the left side is a perfect square...
