The problem asks to multiply and simplify the expression $(5\sqrt{7} + \sqrt{11})(\sqrt{7} - 4\sqrt{11})$.

1. Problem Description

The problem asks to multiply and simplify the expression $(5\sqrt{7} + \sqrt{11})(\sqrt{7} - 4\sqrt{11})$.

2. Solution Steps

We use the distributive property (FOIL method) to multiply the two binomials:

$(a+b)(c+d) = ac + ad + bc + bd$.

$(5\sqrt{7} + \sqrt{11})(\sqrt{7} - 4\sqrt{11}) = (5\sqrt{7})(\sqrt{7}) + (5\sqrt{7})(-4\sqrt{11}) + (\sqrt{11})(\sqrt{7}) + (\sqrt{11})(-4\sqrt{11})$

$= 5(\sqrt{7})^2 - 20\sqrt{7}\sqrt{11} + \sqrt{11}\sqrt{7} - 4(\sqrt{11})^2$

$= 5(7) - 20\sqrt{77} + \sqrt{77} - 4(11)$

$= 35 - 20\sqrt{77} + \sqrt{77} - 44$

$= 35 - 44 - 20\sqrt{77} + \sqrt{77}$

$= -9 - 19\sqrt{77}$

3. Final Answer

$-9 - 19\sqrt{77}$