An algebra problem by anushka sharma

I'm assuming the question meant: the range of values of $m$ such that the expression is defined for all real values of $x$

For this expression to be defined for all real values of $x$ , we must ensure that the denominator $-4x^2+3x+m$ never equals to $0$ for any real value of $x$ . This means that the equation $-4x^2+3x+m=0$ should not have any real roots.

Therefore, we can use the quadratic determinant to find the range of $m$

$b^2-4ac<0\\ 3^2 - 4(-4)(m) <0\\ 9+16m<0 \implies m<-\dfrac{9}{16}$

Therefore, the answer is $\boxed{m<-\dfrac{9}{16}}$ or $\boxed{\left(-\infty, -\dfrac{9}{16}\right)}$