Once you know $k$, substitute $x=4$ into $h(x)$ to create an equation involving $b^{2}$. Solve this equation to find $b$.

After you have both $k$ and $b$, substitute $x=5$ into $h(x)=k b^{x-2}+7$. Be careful with the exponent $x-2$ and with the final $+7$.

In Desmos, type 12-7 to verify that $h(2)-7=5$, so $k=5$ because $h(2)=k+7$.

Type 27-7 to get $20$, then type 20/5 to confirm $b^2=4$. Finally, type sqrt(4) and note the positive result for $b$ (since $b$ is given as positive).

In Desmos, enter the expression 5*2^3+7 (using the $k$ and $b$ you found). The output is the value of $h(5)$; read that value directly from Desmos.

Substitute $x=2$ into the function

When $x=2$, the exponent becomes $2-2=0$, so $b^{0}=1$:

We are told $h(2)=12$, so

Now we know $k=5$.

Substitute $x=4$ and $k=5$ into the function:

We are told $h(4)=27$, so

Because $b$ is positive, $b=2$.

Now substitute $x=5$, $k=5$, and $b=2$ into the function:

Since $b=2$, we have $b^{3} = 2^{3}=8$, so

So the value of $h(5)$ is 47.