For a linear equation in $x$, when would every real value of $x$ make the equation true? What must be true about the two expressions on each side of the equals sign?

After you expand, you should see that the coefficients of $x$ on both sides already match. Concentrate on making the constant terms equal and solve that resulting equation for $m$.

In Desmos, type (m+2)*(4x-7) on one line and 4*(m+2)*x+9 on another line. When you use the letter m, Desmos will prompt you to create a slider for m; add that slider.

Move the m slider and watch how the two graphs change. You are looking for the value of m where the two graphs lie exactly on top of each other (they are the same line), which corresponds to the equation having infinitely many solutions for $x$; read that value of m from the slider.

A linear equation in $x$ has infinitely many solutions only if, after simplifying, both sides are the same expression in $x$. That means the coefficient of $x$ and the constant term must match on both sides, so the equation is true for every real $x$.

Expand the left side: $(m+2)(4x-7)=4(m+2)x-7(m+2)=4mx+8x-7m-14$. Expand the right side: $4(m+2)x+9=4mx+8x+9$.

After expanding, both sides have the same coefficient of $x$, namely $4m+8$, so the $x$-terms already match. For the expressions to be identical, the constant terms must also be equal, so set $-7m-14=9$.

Solve $-7m-14=9$: add 14 to both sides to get $-7m=23$, then divide by $-7$ to get $m=-\dfrac{23}{7}$. For this value of $m$, both sides of the equation are the same linear expression in $x$, so the equation has infinitely many real solutions for $x$.