The mirror equation is given the

$$
begin{align*}
frac{1}{d_{text{o}}} + frac{1}{d_{text{i}}} = frac{1}{f}
end{align*}
$$

where $d_{text{o}}$ is the object distance from the mirror, $d_{text{i}}$ is the image distance, and $f$ is the focal length of the mirror.

We note here that the image is formed behind the mirror, so $d_{text{o}}$ is negative.

The problem requires to solve for the focal length of the mirror. So we plug in the following known values: $d_{text{o}} = 5.9;text{cm}$ and $d_{text{i}} = -3.7;text{cm}$. And then we take the reciprocal.

$$
begin{align*}
frac{1}{f} &= frac{1}{5.9;text{cm}} + frac{1}{-3.7;text{cm}} \ f &= boxed{-9.9;text{cm}}
end{align*}
$$

$f = -9.9$ cm

The mirror equation is given the

$$
begin{align*}
frac{1}{d_{text{o}}} + frac{1}{d_{text{i}}} = frac{1}{f}
end{align*}
$$

where $d_{text{o}}$ is the object distance from the mirror, $d_{text{i}}$ is the image distance, and $f$ is the focal length of the mirror.

We note here that the image is formed behind the mirror, so $d_{text{o}}$ is negative.

The problem requires to solve for the distance of the object from the mirror. We begin by isolating the term $frac{1}{d_{text{o}}}$ on one side of the equation.

$$
begin{align*}
frac{1}{d_{text{o}}} = frac{1}{f} – frac{1}{d_{text{i}}}
end{align*}
$$

We plug in the following known values: $f = -9.6;text{cm}$ and $d_{text{i}} = -4.5;text{cm}$. And then we take the reciprocal.

$$
begin{align*}
frac{1}{d_{text{o}}} &= frac{1}{-9.6;text{cm}} – left( frac{1}{-4.5;text{cm}} right) \ d_{text{o}} &= boxed{8.5;text{cm}}
end{align*}
$$

$d_{text{o}} = 8.5$ cm