In x2+7x+449, the discriminant is 72−4⋅1⋅(449)=49−164949=492−16249=(49−16)(49+16)49=33×6549=214549, though positive, is not the square of a rational number. Hence we cannot factorize it by splitting middle term.
Hence, the way is to find out zeros of quadratic trinomial x2+7x+449. Zeros of ax2+bx+c are given by quadratic formula −b±√b2−4ac2a.
So its zeros, which are two conjugate irrational numbers are given by quadratic formula and are
−7±√2145492 or
−7±√214572 or
−72±√214514 i.e. −72−√214514 and −72+√214514
Now, if α and β are zeros of quadratic polynomial, then its factors are (x−α)(x−β)
Hence factors of x2+7x+449 are (x+72+√214514) and (x+72−√214514) and
x2+7x+449=(x+72+√214514)(x+72−√214514)