When you have to factorize x^2+10x+21, factorize constant term 21 in two parts whose sum is the coefficient of middle term i.e. 10.
It is quite apparent that these factors are 3 and 7 and hence
x^2+10x+21=x^2+3x+color(red)7x+21
= x(x+3)+7(x+3)
= (x+3)(x+7)
Hence missing term in the expression given in question is 7.
Additional Information - This was possible because coefficient of x^2 was 1. In case it is not 1 say the quadratic polynomial is of the type ax^2+bx+c, then we split axxc in two parts whose sum is b. For example, if the polynomial is 3x^2+10x+8, then divide middle term 11 in two parts whose sum is 3xx8=24. These are 4 and 6 and we have
3x^2+11x+8=3x^2+4x+6x+8
= x(3x+4)+2(3x+4)=(x+2)(3x+4)
Note - If signs of a and c are different then divide axxc in two parts whose difference is b.
