x² + 24x + __

Question

what is the answer

Answer 1

(x+a)²=x²+2ax+a² for a perfect square. So 2ax=24x, making a=12, a²=144.

x²+24x+144 is one solution, equivalent to (x+12)².

(x+a)(x+b)=x²+(a+b)x+ab.

If a+b=24, then b=24-a and a can take any value. If ab is positive then 0<a<24.

For example, if a=10 then b=14 and we have x²+24x+140=(x+10)(x+14);

if a=5, b=19 and we have x²+24x+95=(x+5)(x+19).

etc.