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$\begin{array}{l}1)\,\,6x^2 – 7xy – 13y^2\\ = 6x^2 + 6xy – 13xy – 13y^2\\ = 6x(x + y) – 13y(x + y)\\ = (x+y)(6x – 13y)\\ 2)\,\,x^2 – 12xy + 35y^2\\ = x^2 – 5xy – 7xy + 35y^2\\ = x(x – 5y) – 7y(x – 5y)\\ = (x-5y)(x-7y)\\ 3)\,\,50xy – 7x^2 – 7y^2\\ = 49xy – 7x^2 + xy – 7y^2\\ = 7x(7y – x) + y(x – 7y)\\ = (x – 7y)(y – 7x)\\ 4)\,\,2021x^2 – 2020x – 4041\\ = 2021x^2 -2021 – 2020x – 2020\\ = 2021(x^2 – 1) – 2020(x-1)\\ = 2021(x-1)(x+1) – 2020(x-1)\\ = (x-1)[2021(x+1) – 2020]\\ = (x-1)(2021x +1)\\ 5)\,\,4x^2 – y^2 – 12x + 8y – 7\\ = (4x^2 – 12x + 9) – (y^2 – 8y + 16)\\ = (2x – 3)^2 – (y – 4)^2\\ = (2x – 3 + y – 4)(2x – 3 – y + 4)\\ = (2x + y – 7)(2x – y + 1)\\ 6)\,\,9(x^2 – y^2) – 6(2x + 3y) – 5\\ = 9(x – y)(x + y) – 12x – 18y – 5\\ = (3x – 3y)(3x + 3y) + 3x – 3y – 15x – 15y – 5\\ = (3x – 3y)(3x + 3y) + (3x – 3y) – 5(3x + 3y) – 5\\ = (3x – 3y)(3x + 3y + 1) – 5(3x + 3y + 1)\\ = (3x + 3y + 1)(3x – 3y – 5)\end{array}$
$a)6x^2-7xy-13y^2_{}$
$⇔6x^2+6xy-13xy-13y^2_{}$
$⇔6x(x+y)-13y(x+y)_{}$
$⇔(x+y)(6x-13y)_{}$
$b)x^2-12xy+35y^2_{}$
$⇔x^2-5xy-7xy+35y^2_{}$
$⇔x(x-5y)-7y(x-5y)_{}$
$⇔(x-5y)(x-7y)_{}$
$c)50xy-7x^2-7y^2_{}$
$⇔49xy-7x^2+xy-7y^2_{}$
$⇔7x(7y-x)+y(x-7y)_{}$
$⇔(x-7y)(y-7x)_{}$
$d)2021x^2-2020x-4041_{}$
$⇔2021x^2-2021-2020x-2020_{}$
$⇔2021(x62-1)-2020(x-1)_{}$
$⇔2021(x-1)(x+1)-2020(x-1)_{}$
$⇔(x-1). [ 2021(x+1)-2020]_{}$
$⇔(x-1)(2021x+1)_{}$
$e)4x^2-y^2-12x+8y-7_{}$
$⇔(4x^2-12x+9)-(y^2-8y+16)_{}$
$⇔(2x-3)^2-(y-4)^2_{}$
$⇔(2x-3+y-4)(2x-3-y+4)_{}$
$⇔(2x+y-7)(2x-y+1)_{}$
$f)9(x62-y^2)-6(2x+3y)-5_{}$
$⇔9(x-y)(x+y)-12x-18y-5_{}$
$⇔(3x-3y)(3x+3y)+3x-3y-15x-15y-5_{}$
$⇔(3x-3y)(3x+3y)+(3x-3y)-5(3x+3y)-5_{}$
$⇔(3x-3y)(3x+3y+1)-5(3x+3y+1)_{}$
$⇔(3x+3y+1)(3x-3y-5)_{}$