What value of $m$ ensures that the two roots of the quadratic equation $4x^2+4(3m-1)x+(m+7)=0$ are reciprocal to each other?

For a general quadratic equation $a{x}^2+bx+c=0$, if the roots are reciprocal to each other (i.e., $\alpha$ and $\frac{1}{\alpha }$), their product must be $1$:
$$\text{Product of roots}=\frac{c}{a}=1⟹c=a$$

Comparing the given equation $4x^2+4(3m-1)x+(m+7)=0$ to the general form:

* $a=4$
* $c=m+7$

Equating $c=a$:

$$m+7=4⟹m=4-7=-3$$