Find coefficient of $x^2$ Let $C$ be the coefficient of $x^2$ in the expansion of the product
$$(1-x)(1+2x)(1-3x) \cdots (1+14x)(1-15x)$$
Find: $\lvert C \rvert $

Multiply the monomials one by one and discard the terms of degree above $2$:
$$1-x\\1+2x\to1+ x-2 x^2\\
1 -3 x\to1 -2 x-5 x^2\\
1+ 4 x\to1+ 2 x-13 x^2\\
1 -5 x\to1 -3 x-23 x^2\\
1+ 6 x\to1+ 3 x-41 x^2\\
1 -7 x\to1 -4 x-62 x^2\\
1+ 8 x\to1+ 4 x-94 x^2\\
1 -9 x\to1 -5 x-130 x^2\\
1+ 10 x\to1+ 5 x-180 x^2\\
1 -11 x\to1 -6 x-235 x^2\\
1+ 12 x\to1+ 6 x-307 x^2\\
1 -13 x\to1 -7 x-385 x^2\\
1+ 14 x\to1+ 7 x-483 x^2\\
1 -15 x\to1 -8 x\color{green}{-588} x^2\\$$
Notice that the terms of the first degree follow a simple pattern and that the final result is a sum of pairwise products.
$$0-2\cdot1-3\cdot1-4\cdot2-5\cdot2-\cdots15\cdot7$$

$$(1+ax+bx^2)(1+cx)\to1+(a+c)x+(b+ac)x^2$$
