Answer:
[tex]\sf (1) \: \left(x-2y\right)\left(3x+5y-1\right)-x\left(x+y\right)[/tex]
Apply Distributive property by multiplying term x-2y by the terms of 3x+5-1:
[tex]\sf 3x^2-xy-x-10y^2+2y-x\left(x+y\right)[/tex]
Multiply x by x+y:
[tex]\sf 3x^2-xy-x-10y^2+2y-x^2-xy[/tex]
Combine like terms:
[tex]\sf (3x^2+(-x^2))[/tex] [tex]\sf (-xy+(-xy))[/tex]
↓
[tex]\sf 2x^2[/tex] [tex]\sf -2xy[/tex]
[tex]\mapsto \boxed{\sf 2x^2-2xy-x-10y^2+2y}[/tex]
_____________________________
[tex]\sf(2) \left(b-1\right)\left(b^3-2b^2+3b-4\right)-\left(2b-3\right)[/tex]
Apply Distributive property, Multiply b -1 by b^3-2b^2+3b-4, and then combine like terms:
[tex]\boxed{\sf -\left(a-b\right)=-a+b}[/tex]
[tex]\sf \left(b-1\right)\left(b^3-2b^2+3b-4\right)-2b+3[/tex]
[tex]\sf b^4-3b^3+5b^2-7b+4-2b+3[/tex]
Combine -7b + (-2b) = -9b:
[tex]\sf b^4-3b^3+5b^2-9b+4+3[/tex]
Add 4 and 3:
[tex]\mapsto \boxed{\sf b^4-3b^3+5b^2-9b+7}[/tex]
___________________________
[tex]\sf (3)\:\left(-x^2-y^2+z^2\right)+\left(x-y\right)\left(x+y\right)[/tex]
**Apply Rule: →
[tex]\boxed{\sf (a-b)(a+b)=a^2-b^2}[/tex]
[tex]\sf -x^2-y^2+z^2+x^2-y^2[/tex]
Combine -y^2 and -y^2 = -2y, and -x^2+ x^2 = 0
[tex]\mapsto \sf \boxed{z^2-2y^2}[/tex]
____________________________
[tex]\sf (4) (3x+7a)(3x-7a)(2x+a)(2x+3a)[/tex]
*Rule:
[tex]\sf (a-b)(a+b)=a^2-b^2[/tex]
[tex]\sf (3x)^2-(7a)^2+(2x+a)(2x+3a)[/tex]
↓ ↓
Calculate:
[tex]\sf 9x^2-49a^2+(2x+a)(2x+3a)[/tex]
Apply Distributive property:
Multiply 2x+a by 2x+3a:
[tex]\sf 9x^2-49^2+4x^2+6xa+2ax+3a^2[/tex]
Combine like terms:
[tex]\mapsto \boxed{\sf13x^2-46a^2+8xa}[/tex]
_____________________________
