In the \(xy\)-plane, the graph of   \(y=x^2+2x+a\)   contains the points \((-1,0 )\) and \( (2,b) \). What is the value of \(b\)?

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We first substitute the point with known \(x\) and \(y\) coordinates to find the value of \(a\):

$$ y=x^2+2x+a $$ $$ 0=(-1)^2+2(-1)+a $$ $$ 0=1-2+a $$ $$ 0=-1+a$$ $$1=a$$ We obtain the full equation and use the other point to find \(b\). $$ y=x^2+2x+1 $$ $$ b=(2)^2+2(2)+1$$ $$ b=4+4+1$$ $$ b=\boxed{9}$$