First, factor the coefficient of x2 out of the first two terms to get −2x2−7x+4=−2(x2+72x)+4=0.
Next, take the coefficient of x inside the parentheses, 72, divide it by 2 to get 74, and then square that number to get 4916. Add this number inside the parentheses and then "balance" it by adding −2⋅4916 on the other side of the equation to get −2(x2+72x+4916)+4=−2⋅4916=−498.
The reason this trick is a good idea is that the expression x2+72x+4916 is a perfect square. It equals (x+74)2, so the equation becomes −2(x+74)2+4=−498, which is equivalent to −2(x+74)2=−818 and (x+74)2=8116.
Now take the ± square root of both sides to get x+74=±94, leading to two solutions x=94−74=24=12 and x=−94−74=−164=−4.
You should check these in the original equation:
x=12⇒−2(12)2−7(12)+4=−12−72+4=−4+4=0
x=−4⇒−2(−4)2−7(−4)+4=−32+28+4=0
