Answer
$x$ = $-4$ and $-2$
Work Step by Step
Given equation is-
$\frac{4}{x+2}-\frac{6}{2x}$ = $\frac{5}{2x+4}$
i.e. $\frac{8x-6(x+2)}{2x(x+2)}$ = $\frac{5}{2x+4}$
i.e. $\frac{8x-6x-12}{2x(x+2)}$ = $\frac{5}{2x+4}$
i.e. $\frac{2x-12}{2x(x+2)}$ = $\frac{5}{2x+4}$
i.e. $\frac{2(x-6)}{2x(x+2)}$ = $\frac{5}{2x+4}$
i.e. $\frac{x-6}{x(x+2)}$ = $\frac{5}{2x+4}$
i.e. $(x-6)(2x+4)$ = $5x(x+2)$
i.e. $2x^{2}+4x-12x-24$ = $5x^{2}+10x$
i.e. $5x^{2}+10x -2 x^{2}+8x +24$ = $0$
i.e. $3x^{2}+18x +24$ = $0$
Algebraic Solution:
$3x^{2}+18x +24$ = $0$
i.e. $3x^{2}+12x+6x +24$ = $0$
i.e. $3x(x+4)+6(x +4)$ = $0$
i.e. $(x+4)(3x +6)$ = $0$
if $(x+4)$ = $0$, then $x$ = $-4$
if $(3x+6)$ = $0$, then $x$ = $-\frac{6}{3}$ = $-2$
i.e. $x$ = $-4$ and $-2$ are required solutions
Graphical Solution:
$3x^{2}+18x +24$ = $0$
Now graphing $y$ = $3x^{2}+18x +24$
$x-intercepts$ are $-4$ and $-2$
i.e. $x$ = $-4$ and $-2$ are required solutions
