当 $f(x)$ 除以 $x - 1$ 时，余数为 $61$。 当 $f(x)$ 除以 $x + 3$ 时，余数为 $-123$。 求 $f(x)$ 除以 $x^2 + 2x - 3$ 时的余式。

1. Express $f(x)$ in terms of the divisor, quotient, and a linear remainder

Since the divisor $x^2 + 2x - 3$ is a quadratic polynomial, the remainder will be at most a linear polynomial of the form $ax + b$.
Let $f(x) = Q(x)(x^2 + 2x - 3) + ax + b$
$f(x) = Q(x)(x - 1)(x + 3) + ax + b$

2. Use the Remainder Theorem to form simultaneous equations

By the Remainder Theorem:
$f(1) = 61 \implies a(1) + b = 61 \implies a + b = 61$ --- (1)
$f(-3) = -123 \implies a(-3) + b = -123 \implies -3a + b = -123$ --- (2)

3. Solve the simultaneous equations for $a$ and $b$

Subtracting (2) from (1):
$(a + b) - (-3a + b) = 61 - (-123)$
$4a = 184$
$a = 46$

Substitute $a = 46$ into (1):
$46 + b = 61$
$b = 15$

4. State the final remainder

The remainder is $46x + 15$.