Question 1204811
<pre>
Fill in the blanks with positive integers:
(3 + sqrt(5))*3*(5 + 3*sqrt(5))^3 = ___ + ___* sqrt(5)

{{{matrix(1,3, (3 + sqrt(5))*3*(5 + 3*sqrt(5))^3, "= ___ + ___*", sqrt(5))}}}
{{{matrix(1,5, (3 + sqrt(5))*3*(5 + 3*sqrt(5))^3, "=", 3(3 + sqrt(5))(5 + 3*sqrt(5))^3,
"=", (9 + 3sqrt(5))(5 + 3*sqrt(5))^3)}}}

The cube of the sum of a binomial, or {{{(a + b)^3}}} = {{{a^3 + 3a^2b + 3ab^2 + b^3}}}. 
                          As such, {{{matrix(1,3, (5 + 3*sqrt(5))^3, "=", 5^3 + 3(5^2)*3sqrt(5) + 3(5)(3sqrt(5))^2 + (3sqrt(5))^3)}}}
                                             {{{matrix(1,2, "=", 125 + 225sqrt(5) + 15(9 * 5) + 27(sqrt(5))^3)}}}
                                             {{{matrix(1,2, "=", 125 + 225sqrt(5) + 675 + 27 * 5sqrt(5))}}}
                                             {{{matrix(1,2, "=", 800 + 225sqrt(5) + 135sqrt(5))}}}
                                             {{{matrix(1,2, "=", 800 + 360sqrt(5))}}}

We now see that: {{{matrix(1,3, (9 + 3sqrt(5))(5 + 3*sqrt(5))^3, "=", 
(9 + 3sqrt(5))(800 + 360sqrt(5)))}}}. FOILing this gives us:
                                      {{{9(800) + 9(360sqrt(5)) + 800(3sqrt(5)) + (3sqrt(5))(360sqrt(5))}}}
                                      {{{"7,200" + "3,240"sqrt(5) + "2,400"sqrt(5) + ("1,080" * 5)}}}
                                      {{{"7,200" + "3,240"sqrt(5) + "2,400"sqrt(5) + "5,400"}}}
                                      {{{"12,600" + "5,640"sqrt(5)}}}

                       Therefore, {{{matrix(1,3, highlight(highlight_green(highlight((3 + sqrt(5))*3*(5 + 3*sqrt(5))^3))), "=", highlight(highlight_green(highlight("12,600" + "5,640"sqrt(5))))))}}}</pre>