Question 76987
Call the 2 opposite sides {{{x}}}
and the side parallel to the garage {{{150 - 2x}}}
Note that {{{x + x + (150 - 2x) = 150}}}
Call the  area A
{{{A = x(150 - 2x)}}}
{{{A = -2x^2 + 150x}}}
If the coefficient is negative, that means this
parabola has a maximum
When the form of the equation is {{{y = ax^2 + bx + c}}},
The maximum occurs at {{{x = -b / 2a}}}
where {{{a = -2}}}, {{{b = 150}}}
{{{-b / 2a = 150 / 4}}}
{{{150 / 4 = 37.5}}}
When x = 37.5 feet, the area is a maximum
check answer
Does the perimeter equal 150 feet?
{{{x + x + (150 - 2x) = 150}}}
{{{2*37.5 + 75 = 150}}}
{{{75 + 75 = 150}}}
OK
Is the area a little less if I make x a little 
smaller(like 37) or a little bigger (like 38)
I'lll leave this checking for you