Answer
$b$ is not a linear combination of $a_1$, $a_2$, and $a_3$
Work Step by Step
Asking if $b$ is a linear combination of those vectors is equivalent to asking whether the system whose augmented matrix is shown below is consistent.
$$
\begin{bmatrix}
1 & 0 & 2 & -5 \\
-2 & 5 & 0 & 11 \\
2 & 5 & 8 & -7
\end{bmatrix}
$$
We can determine this through row reduction. First, add the second row to the third:
$$
\begin{bmatrix}
1 & 0 & 2 & -5 \\
-2 & 5 & 0 & 11 \\
0 & 10 & 8 & 4
\end{bmatrix}
$$
Now add twice the first row to the second:
$$
\begin{bmatrix}
1 & 0 & 2 & -5 \\
0 & 5 & 4 & 1 \\
0 & 10 & 8 & 4
\end{bmatrix}
$$
Double the second row and subtract it from the third:
$$
\begin{bmatrix}
1 & 0 & 2 & -5 \\
0 & 5 & 4 & 1 \\
0 & 0 & 0 & 2
\end{bmatrix}
$$
The last row is mathematically impossible ($0=2$), so the matrix is inconsistent and $b$ is not a linear combination of the vectors.
