Without using the calculator, find the values of the integer $a$ and $b$ such that $\frac{4 - \sqrt{3}}{a + b\sqrt{3}} = \frac{2 + \sqrt{3}}{4 - \sqrt{3}}.$

1. Cross-multiply to remove fractions

$(4 - \sqrt{3})^2 = (a + b\sqrt{3})(2 + \sqrt{3})$

2. Expand both sides of the equation

$16 - 8\sqrt{3} + 3 = 2a + a\sqrt{3} + 2b\sqrt{3} + 3b$
$19 - 8\sqrt{3} = (2a + 3b) + (a + 2b)\sqrt{3}$

3. Equate rational and irrational parts to form simultaneous equations

$2a + 3b = 19 \quad \text{--- (1)}$
$a + 2b = -8 \quad \text{--- (2)}$

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

From (2), $a = -8 - 2b$.
Substitute into (1):
$2(-8 - 2b) + 3b = 19$
$-16 - 4b + 3b = 19$
$-b = 35 \implies b = -35$
Substitute $b = -35$ into $a = -8 - 2b$:
$a = -8 - 2(-35) = -8 + 70 = 62$