$x^2+xy+zx+yz$

= x(x + y) + z(x + y) = (x + y)(x + z) .

$a^2+bc+ca+ab$

= $a^2+bc+ca+ab$

= a(a + b) + c(a + b) =(a+b)(a+.c).

$b(px+qy)+a^{2}qx+b^{2}py$

$= abpx + abqy + a^{2}qx + b^{2}py$

$= abpx + a^{2}qx + b^{2}py + abqy$

= ax(bp + aq) + by(bp + aq)

= (bp + aq)(ax + by)

= (ax + by)(bp + aq)

$4x^2-y^2$

$=(2x{)}^2-(y{)}^2=(2x+y)(2x-y).$

$9a^2-4b^2$

$=(3a{)}^2-(2b{)}^2=(3a+2b)(3a-2b).$

$a^2{b}^2-49y^2$

$=(ab{)}^2-(7y{)}^2$

= (ab + 7y)(ab - 7y)

$16x^4-81y^4$

$=(4x^2{)}^2-(9y^2{)}^2$

$=(4x^2+9y^2)(4x^2-9y^2)$

$=(4x^2+9y^2)\left\{\right(2x{)}^2-\left(3y{)}^2\right\}$

$=(4x^2+9y^2)(2x+3y)(2x-3y)$

$=(2x+3y)(2x-3y)(4x^2+9y^2).$

$a^2-(x+y{)}^2$

$=\{a+(x+y\left)\right\}\{a-(x+y\left)\right\}$

= (a + x + y)(a - x - y)

$(2x-3y+5z{)}^2-(x-2y+3z{)}^2$

= {(2x - 3y + 5z) + (x - 2y + 3z)}{(2x - 3y + 5z) - (x - 2y + 3z)}

= (2x - 3y + 5z + x - 2y + 3z)(2x - 3y + 5z - x + 2y - 3z)

= (3x - 5y + 8z)(x - y + 2z) .

$4+8a^2+9a^4$

$=9a^4+8a^2+4$

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

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

$=(3a^2+2+2a)(3a^2+2-2a).$

$=(3a^2+2a+2)(3a^2-2a+2)$

$2a^2+6a-80$

$=2(a^2+3a-40)$

$=2(a^2+8a-5a-40)$

= 2{a(a + 8) - 5(a + 8)} = 2(a + 8)(a - 5) .

$y^2-6y-91$

$=y^2-13y+7y-91$

= y(y - 13) + 7(y - 13)

= (y - 13)(y + 7) .

$p^2-15p+56$

$=p^2-8p-7p+56$

= p(p - 8) - 7(p - 8)

= (p - 8)(p - 7)

$45a^8-5a^4{x}^4$

$=5a^4 (9a^4-x^4)$

$=5a^4 \left\{\right(3a^2{)}^2-\left(x^2{)}^2\right\}$

$=5a^4 (3a^2+x^2)(3a^2-x^2).$

$a^2+3a-40$

$=a^2+8a-5a-40$

$= a(a + 8) - 5(a + 8)$

= (a + 8)(a - 5) .

$(x^2+1{)}^2-(y^2+1{)}^2$

$=\left\{\right(x^2+1)+(y^2+1\left)\right\}\left\{\right(x^2+1)-(y^2+1\left)\right\}$

$=(x^2+1+y^2+1)(x^2+1-y^2-1)$

$=(x^2+y^2+2)(x^2-y^2)$

$=(x^2+y^2+2)(x+y)(x-y).$

$x^2+11x+30$

$=x^2+5x+6x+30$

= x(x + 5) + 6(x + 5)

= (x + 5) . (x+6).

$a^2-b^2+2bc-c^2$

$=a^2-(b^2-2bc+c^2)$

$=a^2-(b-c{)}^2$

= {a + (b - c)}{a - (b - c)}

= (a + b - c)(a - b + c)

$144x^7-25x^3 a^4$

$=x^3 (144x^4-25a^4)$

$=x^3 \left\{\right(12x^2{)}^2-\left(5a^2{)}^2\right\}$

$=x^3 (12x^2+5a^2)(12x^2-5a^2)$

$4x^2+12xy+9y^2-16a^2$

$=(2x{)}^2+2\times 2x\times 3y+(3y{)}^2-(4a{)}^2$

$=(2x+3y{)}^2-(4a{)}^2$

$= (2x + 3y + 4a)(2x + 3y - 4a) .$