i) x(x+y)-3x-3y j) x(x+y)-7x-7y k) x^2(x-y)+2x-2y 1) x(x+3y)-5x-15 m) x^2+3x-x-3 n) x^2-xy+x-y 0) xy+y^2-x-y p) xy+y-2x-2 q) x^3-2x^2+x-2 r) x^4+x^3y-x-y s) x^2+xy-xz-yz t) xy+xz+3y+3z u) x^2-3x+xy-3y v) xy-3x-y^2+3y w) x^2+2x-xy-2y x) 3x^2-x-3xy+y

This question is about factoring algebraic expressions. Factoring is the process of breaking down an expression into its simplest terms (the factors) that when multiplied together give the original expression. This is a fundamental skill in algebra and is used to simplify complex expressions, solve equations, and understand the properties of numbers and functions. <br /><br />For each of the given expressions, we will look for common factors and use the distributive property to factor them. The distributive property states that for all real numbers a, b, and c, a(b + c) = ab + ac and a(b - c) = ab - ac. <br /><br />Let's go through each expression:<br /><br />i) $x(x+y)-3x-3y$ = $x(x+y)-3(x+y)$ = $(x-3)(x+y)$<br />j) $x(x+y)-7x-7y$ = $x(x+y)-7(x+y)$ = $(x-7)(x+y)$<br />k) $x^{2}(x-y)+2x-2y$ = $x^{2}(x-y)+2(x-y)$ = $(x^2+2)(x-y)$<br />l) $x(x+3y)-5x-15$ = $x(x+3y)-5(x+3y)$ = $(x-5)(x+3y)$<br />m) $x^{2}+3x-x-3$ = $x(x+3)-1(x+3)$ = $(x-1)(x+3)$<br />n) $x^{2}-xy+x-y$ = $x(x-y)+1(x-y)$ = $(x+1)(x-y)$<br />o) $xy+y^{2}-x-y$ = $y(x+y)-1(x+y)$ = $(y-1)(x+y)$<br />p) $xy+y-2x-2$ = $y(x+1)-2(x+1)$ = $(y-2)(x+1)$<br />q) $x^{3}-2x^{2}+x-2$ = $x^2(x-2)+1(x-2)$ = $(x^2+1)(x-2)$<br />r) $x^{4}+x^{3}y-x-y$ = $x^3(x+y)-1(x+y)$ = $(x^3-1)(x+y)$<br />s) $x^{2}+xy-xz-yz$ = $x(x+y)-z(x+y)$ = $(x-z)(x+y)$<br />t) $xy+xz+3y+3z$ = $x(y+z)+3(y+z)$ = $(x+3)(y+z)$<br />u) $x^{2}-3x+xy-3y$ = $x(x-3)+y(x-3)$ = $(x+y)(x-3)$<br />v) $xy-3x-y^{2}+3y$ = $y(x-3)-1(x-3)$ = $(y-1)(x-3)$<br />w) $x^{2}+2x-xy-2y$ = $x(x+2)-y(x+2)$ = $(x-y)(x+2)$<br />x) $3x^{2}-x-3xy+y$ = $3x(x-y)+1(x-y)$ = $(3x+1)(x-y)$<br /><br />In each case, we have factored the expression into the product of two binomials. This process can be repeated for more complex expressions, and it is a valuable tool in algebraic manipulation and problem solving.