Problem

Determine whether \(y=(x+1)^{2}+4\) and \(y=1\) intersect in the real plane. Reduce the equality to a squared expression, justify the classification, and explain how the displayed answer graph confirms it.

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set equal=(x+1)²+4=1; reduced=(x+1)²=-3; claimed branch=x+1=-3 so x=-4; classification=one real intersection; graph meaning=line meets parabola once

set equal=(x+1)²+4=1; reduced=(x+1)²=-3; claimed roots=x=-1±sqrt(3); classification=two real intersections; graph meaning=line y=1 crosses parabola twice

set equal=(x+1)²+4=1; reduced=(x+1)²=0; root=x=-1; classification=one tangent intersection; graph meaning=line y=1 touches parabola at its vertex

set equal=(x+1)²+4=1; reduced=(x+1)²=-3; real-solution test=a real square cannot be negative; classification=no real solution; graph meaning=line y=1 lies below parabola whose minimum is4

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Determine whether \(y=(x+1)^{2}+4\) and \(y=1\) intersect in the real plane. Reduce the equality to a squared expression, justify the classification, and explain how the displayed answer graph confirms it.

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