<s>[INST] value of $a$ for which $25^x+(a+2)5^x-(a+3)<0$for at least one real $x$ 
Find the values of $a$ for which the inequality is satisfied for $25^x+(a+2)5^x-(a+3)<0$
for at least one real value of $x$

$\bf{My\; Try::}$ We can write it as $a(5^x-1)<-\left[25^x+2\cdot 5^x-3\right]$
So $$a < -\left(\frac{25^x-5^x+3\cdot 5^x-3}{5^x-1}\right) = -\frac{(5^x+3)(5^x-1)}{5^x-1} = -(5^x+3)$$
Now how can i solve after that help required, Thanks
 [/INST] Let $$(5^x)^2+(a+2)5^x-(a+3)=-y$$
As $5^x$ is real, we need the discriminant  $$(a+2)^2-4(y-a-3)\ge0$$
$$4y\le a^2+8a+16=(a+4)^2$$
Now we need $y>0$ which will be true if $(a+4)^2>0$ which is true for $a+4\ne0\iff a\ne-4$
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