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Answer:
a. 4x^2 +16x -48 = 4x^2 +16x -48
b. -3x^2 +12x +36 = -3x^2 +12x +36
c. x^2 +12x +35 = x^2 +12x +35
Step-by-step explanation:
In general, you would prove this by transforming one of the expressions into the other. The easiest would be to transform the factored form into the vertex form, perhaps, as this would spare you trying to explain the magic of factoring.
Alternatively, you can transform both expressions into the same (standard) form. I believe that will be the easiest of all.
The product of two binomials is ...
(x +a)(x +b) = x(x +b) +a(x +b) = x^2 +bx +ax +ab
(x +a)(x +b) = x^2 +(a+b)x +ab . . . . after collecting terms
The square of a binomial is the same thing, but with b=a, so ...
(x +a)^2 = x^2 +2a +a^2
Using these forms, we can avoid showing all of the intermediate "work" of making the desired transformations.
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a. 4(x -2)(x +6) = 4(x +2)^2 -64
4(x^2 +4x -12) = 4(x^2 +4x +4) -64 . . . . . expanding the products
4x^2 +16x -48 = 4x^2 +16x +16 -64 . . . . using the distributive property
4x^2 +16x -48 = 4x^2 +16x -48 . . . . . collect terms; expressions are equal
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b. -3(x +2)(x -6) = -3(x -2)^2 +48
-3(x^2 -4x -12) = -3(x^2 -4x +4) +48
-3x^2 +12x +36 = -3x^2 +12x -12 +48
-3x^2 +12x +36 = -3x^2 +12x +36
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c. (x +5)(x +7) = (x +6)^2 -1
x^2 +12x +35 = x^2 +12x +36 -1
x^2 +12x +35 = x^2 +12x +35
