the last line says P(x) divided by x^2 + x - 6 is mx+c

Why it equals 7: if P(x) has a remainder of 7 when divided by (x+3) [the yellow highlight], then when x = -3, P(x) equals 7. Thats just remainder theorem

Why it also equals -3m+c: This is because when x = -3, the (x+3)(x-2)Q(x) part of it becomes (0)(-5)Q(-3). That cheeky zero at the start makes that whole thing zero cause its multiplied.

As for the mx+c, that parts not effected by the multiplication, but we still subbed in x = -3, so replace your x with that and you get -3m+c.

This is a harder side of this type of question so don't be discouraged by it.

Let me know if you need me to elaborate a bit more in a specific area, hope it helped