Answer
$h=-17$
Work Step by Step
For $b$ to be in the plane spanned by $a_1$ and $a_2$, the following matrix must be consistent:
$$
\begin{bmatrix}
1 & -2 & 4 \\
4 & -3 & 1 \\
-2 & 7 & h
\end{bmatrix}
$$
To determine this the conditions for consistency, we row reduce. First, add $-4$ times the first row to the second row:
$$
\begin{bmatrix}
1 & -2 & 4 \\
0 & 5 & -15 \\
-2 & 7 & h
\end{bmatrix}
$$
Then add twice the first row to the third row:
$$
\begin{bmatrix}
1 & -2 & 4 \\
0 & 5 & -15 \\
0 & 3 & h + 8
\end{bmatrix}
$$
And multiply the second row by $3/5$:
$$
\begin{bmatrix}
1 & -2 & 4 \\
0 & 3 & -9 \\
0 & 3 & h + 8
\end{bmatrix}
$$
We can see by inspection that in order for the second and third lines to not reduce to $0=1$, it is necessary that $h+8=-9$.
Therefore, it is necessary that $h=-17$ in order for the system to be consistent and thus for $b$ to be in the span of $\{a_1, a_2\}$.
