Answer
$x = -7$ has multiplicity 1 (odd).
The graph crosses the x-axis.
$x = -2$ has multiplicity 1 (odd).
The graph crosses the x-axis.
$x = 2$ has multiplicity 1 (odd).
The graph crosses the x-axis.
Work Step by Step
Multiplicity and x-Intercepts:
If $r$ is a zero of a polynomial f(x), then $(x-r)^{k}$ is a factor in the full factorization of f.
The exponent $k$ indicates the multiplicity of $r$.
If $r$ is a zero of even multiplicity, then the graph touches the x-axis and turns around at $r$.
If $r$ is a zero of odd multiplicity, then the graph crosses the x-axis at $r$.
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Begin factoring by factoring in pairs...
$f(x)=x^{2}(x+7)-4(x+7)$
... common factor: $(x+7)$...
$f(x)=(x+7)(x^{2}-4)$
... recognize a difference of squares $, x^{2}-2^{2}$
$f(x)=(x+7)(x-2)(x+2)$
Zeros: $-7,-2$ and $2.$
$x = -7$ has multiplicity 1 (odd).
The graph crosses the x-axis.
$x = -2$ has multiplicity 1 (odd).
The graph crosses the x-axis.
$x = 2$ has multiplicity 1 (odd).
The graph crosses the x-axis.
