Question

How do you foil $(- 4x + 5) (- 2x - 3)$?

Answer 1

Hint: We will first use the property that (a + b) (c + d) = a (c + d) + b (c + d). Now, we will use the distributive property to further simplify them and at last we will club the like terms.

Complete step-by-step solution:
We are given that we are required to foil (- 4x + 5) (- 2x - 3).
We know that we have a property given by the following expression:-
$ \Rightarrow $(a + b) (c + d) = a (c + d) + b (c + d)
Now, replacing a by – 4x, b by 5, c by – 2x and d by – 3, we will then obtain the following equation:-
$ \Rightarrow $(- 4x + 5) (- 2x - 3) = - 4x (- 2x – 3) + 5 (- 2x – 3)
Term the above equation as equation number 1.
So, we have (- 4x + 5) (- 2x - 3) = - 4x (- 2x – 3) + 5 (- 2x – 3) …………….(1)
Now, we will first use the distributive property in – 4x (- 2x – 3):
$ \Rightarrow $– 4x (- 2x – 3) = (- 4x) (- 2x) + (- 4x) (- 3)
Simplifying the brackets in the right hand side, we will then obtain the following equation:-
$ \Rightarrow - 4x( - 2x - 3) = 8{x^2} + 12x$ ……………..(2)
Now, we will use the distributive property in 5 (- 2x – 3):
$ \Rightarrow $5 (- 2x – 3) = (5) (- 2x) + (5) (- 3)
Simplifying the brackets in the right hand side, we will then obtain the following equation:-
$ \Rightarrow $5 (- 2x – 3) = - 10 x - 15 ……………..(3)
Putting the equation numbers (2) and (3) in equation number (1), we will then obtain the following equation:-
$ \Rightarrow - 4x( - 2x - 3) + 5( - 2x - 3) = 8{x^2} + 12x - 10x - 15$
We can write the above equation as follows:-
$ \Rightarrow ( - 4x + 5)( - 2x - 3) = 8{x^2} + 12x - 10x - 15$
Clubbing the like terms in the right hand side, we will then obtain the following equation:-
$ \Rightarrow ( - 4x + 5)( - 2x - 3) = 8{x^2} + 2x - 15$

Therefore $8{x^2} + 2x - 15$ is the required answer.

Note: The students must know the definition of the distributive property which we have used in the solution of the above question:-
Distributive Property: It states that for any three numbers a, b and c, we have the following equation with us: $ \Rightarrow $a (b + c) = ab + ac
This is true even if a, b and c are complex numbers and since real numbers are subset of complex numbers, therefore it is true for real numbers as well.