x²+bx+c is the area of a rectangle with width (x+q) and height (x+p). Adjust p and q to build the target polynomial.
When you split the rectangle into 4 pieces, you get x², px, qx, and pq. Add them together and you get x² + (p+q)x + pq. So factoring means finding p and q such that "their sum is b and their product is c."