I5.1 If $a(x+1)=x^{3}+3 x^{2}+3 x+1$, find $a$ in terms of $x$.
I5.2 If $a-1=0$, then the value of $x$ is 0 or $b$, what is $b$ ?
I5.3 If $p c^{4}=32, p c=b^{2}$ and $c$ is positive, what is the value of $c$ ?
I5.4 $P$ is an operation such that $P(A \cdot B)=P(A)+P(B)$.
$P(A)=y$ means $A=10^{y}$. If $d=A \cdot B, P(A)=1$ and $P(B)=c$, find $d$.
$\begin{array}{l}a(x+1)=(x+1)^{3} \\ a=(x+1)^{2}=x^{2}+2 x+1 \\ a=1 \Rightarrow 1=(x+1)^{2} \\ x^{2}+2 x=0 \Rightarrow x=0 \text { or }-2 \Rightarrow b=-2 \\ p c^{4}=32 \ldots \ldots(1) \\ p c=(-2)^{2}=4 \ldots \ldots(2) \\ (1) \div(2): c^{3}=8 \\ c=2 \\ P(A)=1 \Rightarrow A=10^{1}=10 \\ P(B)=c \Rightarrow B=10^{2}=100 \\ d=A \cdot B=10 \cdot 100=1000\end{array}$