What does the curve $x^4-y^2=y+1$ look like?

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The equation rearranges to \[x^4 = y^2 + 2y + 1 = (y+1)^2\] which is true if and only if \[x^2=y+1 \qquad \text{or} \qquad -x^2=y+1.\]

So the graph consists of two parabolae, \(y=x^2-1\) and \(y=-x^2-1\), which both cross the \(y\)-axis at \(y=-1\), and so form the graph in (b).