Number of real values of x satisfying the equation `log_(x^2+6x+8)(log_(2x^2+2x+3)(x^2-2x))=0` is equal to

Explanation

This equatioin is equivalent of the system
`{(x^(2)-6x+8ge0),(x^(2)-6x+8!=1),(2x^(2)-2x-8gt0),(2x^(2)-2x-8!=1),(x^(2)+5x=2x^(2)-2x-8):}`
Solve the equations of this system
`implies{(xlt2 "and"xgt4),(x!=3+-sqrt(2)),(xlt(1-sqrt(17))/2"and"xgt(1+sqrt(17))/2),(x1=(1+-sqrt(19))/2),(x=-1,8):}`
`x=-1`, does not satisfy the third relatiion of this system.
Hence `x_(1)=8` is only rootof the original equation.