Kirchoffs Laws and Class-A Amplifier Circuit
The mistake is in the negative sign of the equation you draw
$$V_{CC}=V_{CE}+\left(-(1+\beta)R_E-\beta R_C\right)i_B$$
If you draw the circuit and plot the currents you get the following diagram



simulate this circuit &ndash; Schematic created using CircuitLab
Then, simply follow the Votage drop, you have Vcc that drops with the flow of the currents throughout the resistors and the jump between collector and emitter as follows:
$$V_{CC}=I_{c}R_{C} + V_{CE} + I_{e}R_{e}$$
You also know that
$$I_{c} = \beta I_{b} $$
$$I_{e} = I_{c} + I_{b} $$
If you replace the currents you get
$$V_{CC}=\beta I_{b}R_{C} + V_{CE} +( \beta I_{b} + I_{b} )R_{e}$$
Making it a bit more readable
$$V_{CC} = V_{CE} + I_{b}( \beta R_{c} + (1+ \beta )R_{e}) $$