Question 515414
To factor {{{112 b^2 + 392 b g + 343 g^2 }}}, we first notice that all of the coefficients are divisible by 7, so we factor 7 out of each term:

{{{ 7(16 b^2 + 56  b g + 49 g^2) }}}

There are no other common factors of the coefficients.  Now we need to find two numbers whose product is (16)(49) = 784 and whose sum is 56.  Let's write out some possible ways to factor 784 and look at the sum of the factors:

(1)(784) sum = 785
(2)(392) sum = 394
(4)(196) sum = 200
(7)(112) sum = 119
(8)(98) sum = 106
(14)(56) sum = 70
(16)(49) sum = 65
(28)(28) sum = 56 <--- this works!

So we've found our combination.  We can rewrite the quadratic expression using these two factors:

{{{7(16 b^2 + 28 b g + 28 b g + 343 g^2)}}}

Now we can factor the quadratic by grouping.  The greatest common factor of {{{16 b^2}}} and {{{28 b g}}} is {{{4b}}}, so we factor that out of the first two terms:

{{{7(4b( 4b + 7g) + 28 b g + 343 g^2)}}}

The greatest common factor of {{{28 b g}}} and {{{343 g^2}}} is {{{7g}}}, so we factor this out of the last two terms:

{{{7(4b(4b + 7g) + 7g(4b + 7g))}}}

Now we factor {{{4b + 7g}}} out of both terms:

{{{7(4b + 7g)(4b + 7g) = 7(4b + 7g)^2}}}

This completes the factorization.