$$\frac{3x+2}{7}+\frac{4(x+1)}{5}=\frac{2}{3}(2x+1)$$
$x=4$
$$15\left(3x+2\right)+21\times 4\left(x+1\right)=70\left(2x+1\right)$$
$$45x+30+21\times 4\left(x+1\right)=70\left(2x+1\right)$$
$$45x+30+84\left(x+1\right)=70\left(2x+1\right)$$
$$45x+30+84x+84=70\left(2x+1\right)$$
$$129x+30+84=70\left(2x+1\right)$$
$$129x+114=70\left(2x+1\right)$$
$$129x+114=140x+70$$
$$129x+114-140x=70$$
$$-11x+114=70$$
$$-11x=70-114$$
$$-11x=-44$$
$$x=\frac{-44}{-11}$$
$$x=4$$
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