Factorise the following quadratic polynomial by splitting the middle term:
$x^{2}-3x-40$

(A) To factorise the quadratic polynomial $x^{2}-3x-40$,we need to find two numbers whose product is $-40$ and whose sum is $-3$.
$1$. The factors of $-40$ are $(1, -40), (-1, 40), (2, -20), (-2, 20), (4, -10), (-4, 10), (5, -8), (-5, 8)$.
$2$. Among these pairs,the pair $(5, -8)$ satisfies the condition: $5 + (-8) = -3$.
$3$. Now,split the middle term $-3x$ as $5x - 8x$:
$x^{2} + 5x - 8x - 40$
$4$. Group the terms and factor out the common factors:
$x(x + 5) - 8(x + 5)$
$5$. Factor out the common binomial $(x + 5)$:
$(x + 5)(x - 8)$