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#.
