$$\frac{a+1}{2}+\frac{3a-2}{5}=\frac{5a+1}{4}$$
$a=-1$
$$10\left(a+1\right)+4\left(3a-2\right)=5\left(5a+1\right)$$
$$10a+10+4\left(3a-2\right)=5\left(5a+1\right)$$
$$10a+10+12a-8=5\left(5a+1\right)$$
$$22a+10-8=5\left(5a+1\right)$$
$$22a+2=5\left(5a+1\right)$$
$$22a+2=25a+5$$
$$22a+2-25a=5$$
$$-3a+2=5$$
$$-3a=5-2$$
$$-3a=3$$
$$a=\frac{3}{-3}$$
$$a=-1$$
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