Question 83666
First you give a name to the two consecutive numbers to make things easier:
If you call the first one x, then the next one is x+1.

If twice the larger is added to four times the square of the smaller, the sum will be 112.

{{{2(x+1) + 4x^2 = 112}}}
{{{2x + 2 + 4x^2 = 112}}}

Subtracting 2 from both sides:
{{{4x^2+2x = 110}}}
You can divide through by 2:
{{{2x^2+x=55}}}
Now rearrange your equation to look something like this:
{{{2x^2+x-55=0}}}
Factorising:
{{{(2x+11)(x-5)=0}}}
So this gives either x=-5.5 or x=5
the first number is rejected so we'll take the second.
So your two numbers are: 5 & 6