Answer
$x=-\dfrac{5\log3-\log4}{\log4+2\log3}\approx-1.15$
Work Step by Step
$4^{1-x}=3^{2x+5}$
Apply $\log$ to both sides of the equation:
$\log4^{1-x}=\log3^{2x+5}$
Take $1-x$ and $2x+5$ down to multiply in front of their respective logarithms:
$(1-x)\log4=(2x+5)\log3$
Evaluate the products:
$\log4-x\log4=2x\log3+5\log3$
Take $2x\log3$ to the left side and $\log4$ to the right side:
$-x\log4-2x\log3=5\log3-\log4$
Take out common factor $x$ from the left side:
$x(-\log4-2\log3)=5\log3-\log4$
Solve for $x$:
$x=-\dfrac{5\log3-\log4}{\log4+2\log3}\approx-1.15$