First, factor the coefficient of x^2 out of the first two terms to get -2x^2-7x+4=-2(x^2+7/2 x)+4=0.
Next, take the coefficient of x inside the parentheses, 7/2, divide it by 2 to get 7/4, and then square that number to get 49/16. Add this number inside the parentheses and then "balance" it by adding -2*49/16 on the other side of the equation to get -2(x^2+7/2 x+ 49/16)+4=-2*49/16=-49/8.
The reason this trick is a good idea is that the expression x^2+7/2 x+ 49/16 is a perfect square. It equals (x+7/4)^2, so the equation becomes -2(x+7/4)^2+4=-49/8, which is equivalent to -2(x+7/4)^2=-81/8 and (x+7/4)^2=81/16.
Now take the \pm square root of both sides to get x+7/4=\pm 9/4, leading to two solutions x=9/4-7/4=2/4=1/2 and x=-9/4-7/4=-16/4=-4.
You should check these in the original equation:
x=1/2\Rightarrow -2(1/2)^2-7(1/2)+4=-1/2-7/2+4=-4+4=0
x=-4\Rightarrow -2(-4)^2-7(-4)+4=-32+28+4=0
