Question 67738
If the problem is to solve for x, which this appears to be, then you are proceeding correctly, except for the results of applying the distributive property.  I shouldn't be too concerned with not having a quadratic equation in the final stages. Solve for x.
{{{(x+12)(x+12)-(x+1)(x+7) = 217}}}
{{{(x^2+24x+144) - (x^2+8x+7) = 217}}} Simplify the left side.
{{{x^2-x^2+24x-8x+144-7 = 217}}} Continue simplifying.
{{{16x+137 = 217}}} Subtract 137 from both sides.
{{{16x = 80}}} Finally, divide both sides by 16.
{{{x = 5}}}

As you can see, the {{{x^2}}}'s drop out so you no longer have a quadratic equation but you can still solve for x.