Question 753021
Giving you some tricky questions are they? I think we can handle it. It just a little complicated on-line.
First, to reduce the order of the polynomial, let
(1) y = x^2, then your equation reduces to
(2) y^2 + sqrt(76)*y -76 = 0
Now use the quadratic equation to find the roots of (2).
(3) y = {{{-sqrt(76)/2 +- (1/2)*sqrt(76 -4*1*(-76))}}} or
(4) y = {{{-sqrt(76)/2 +- (1/2)*sqrt(76 + 4*76)}}} or
(5) y = {{{-sqrt(76)/2 +- (1/2)*sqrt(76)*(sqrt(5))}}} or
(6) y = {{{(sqrt(76)/2)*(-1 +- sqrt(5))}}}
Since y = x^2 we get 
(7) x = +/-sqrt(y) and since y has two roots we will get four roots for x.
Taking the square root of (6) gives us two roots of x
(8) {{{x1 = +sqrt((sqrt(76)/2)*(-1+sqrt(5)))}}}  and
(9) {{{x2 = -sqrt((sqrt(76)/2)*(-1+sqrt(5)))}}}   
Likewise we can write the other two roots of x as 
(10) {{{x3 = +sqrt((sqrt(76)/2)*(-1-sqrt(5)))}}}  and
(11) {{{x4 = -sqrt((sqrt(76)/2)*(-1-sqrt(5)))}}}  
We acn check these roots by substitution into your original equation in x. I'll use x of (8) to show that
(12) {{{x^4 + sqrt(76)*x^2 -76 = 0}}}
We have
(13) {{{x^2 = sqrt(76)/2*(-1+sqrt(5))}}} and
(14) {{{x^4 = 76/4*(-1+sqrt(5))^2}}}
Now put (13) and (14) into (12) and get
(15) {{{76/4*(-1+sqrt(5))^2 + sqrt(76)*sqrt(76)/2*(-1+sqrt(5)) - 76 = 0}}} or
(16) {{{76/4*(-1+sqrt(5))^2 + 76/2*(-1+sqrt(5)) - 76 = 0}}} 
Multiply (16) by 4/76 and get
(17) {{{(-1+sqrt(5))^2 + 2*(-1+sqrt(5)) - 4 = 0}}} or
(18) {{{(-1+sqrt(5))*(-1+sqrt(5)+2) - 4 = 0}}} or
(19) {{{(-1+sqrt(5))*(+1+sqrt(5)) - 4 = 0}}} or
(20) {{{-1+sqrt(5)^2 - 4 = 0}}} or
(21) {{{-1 + 5 - 4 = 0}}} or
(22)  0 = 0 check
Answers: the roots of your quardic equation are given by (8)-(11)