Question

The following steps are used to rewrite the polynomial expression (x+2y)(x-5y)

Answer 1

(x+2y)(x-5y)
do foil method:
x^2-5xy+2xy-10y^2
simplify x^2-3xy-10y^2,
so d is correct.

Answer 2

Answers
Step 1) Distributive property
Step 2) Distributive property
Step 3) Commutative property of multiplication
Step 4) Combine like terms

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The distributive property is the rule
a*(b+c) = a*b+a*c

we can make it a bit more complicated if we replace 'a' with some other expression, say x+y.

So
a*(b+c) = a*b + a*c
turns into
(x+y)*(b+c) = (x+y)*b + (x+y)*c
and then we can use the distributive property again to further distribute the outer letters into each parenthesis grouping
(x+y)*(b+c) = (x+y)*b + (x+y)*c
(x+y)*(b+c) = x*b+y*b + x*c+y*c

This is what the FOIL rule is all about. Though the FOIL rule is limited compared to what the distributive property can do. FOIL only applies to multiplication of two binomials. The distributive property can be applied to multiplying any terms.

The commutative property of multiplication is the idea that you can multiply any two numbers in any order you want
example: 2*3 = 6 and 3*2 = 6
In general: x*y = y*x