Answer
The solutions are $u=\displaystyle \frac{-1+5\sqrt{3}}{2}$ or $u=\displaystyle \frac{-1-5\sqrt{3}}{2}$.
Work Step by Step
$ 4u^{2}+4u+1=75\qquad$ ...write left side as a binomial squared.($(a+b)^{2}=a^{2}+2ab+b^{2}$)
$a=2u,b=1$
$(2u+1)^{2}=75\qquad$ ...take square roots of each side.
$ 2u+1=\pm\sqrt{75}\qquad$ ...add $-1$ to each side.
$ 2u+1-1=\pm\sqrt{75}-1\qquad$ ...simplify.
$ 2u=-1\pm\sqrt{75}\qquad$ ...rewrite $\sqrt{75}$ as$ \sqrt{25}\cdot\sqrt{3}2$
$ 2u=-1\pm\sqrt{25}\cdot\sqrt{3}\qquad$ ...evaluate $\sqrt{25}$.
$ 2u=-1\pm 5\sqrt{3}\qquad$ ...divide both sides with $2$.
$ u=\displaystyle \frac{-1\pm 5\sqrt{3}}{2}\qquad$ ...write as separate equations.
$u=\displaystyle \frac{-1+5\sqrt{3}}{2}$ or $u=\displaystyle \frac{-1-5\sqrt{3}}{2}$
