When multiplying two powers of the same base, you add the exponents:
[tex] a^b\cdot a^c = a^{b+c} [/tex]
So, in your case, you have
[tex] e^x\cdot e^{x+1} = e^{x+(x+1)} = e^{2x+1} [/tex]
Now, an exponential function equals 1 if and only if the exponent equals zero. So, you have
[tex] e^{2x+1} = 1 \iff 2x+1 = 0 \iff 2x = -1 \iff x=\dfrac{-1}{2} [/tex]
