Question

type the three letters with commas in between in the answer box below.
select the three pairs of equivalent expressions.

a ((3x^2 + 7x - 4) + (x^2 + x + 3); 4x^2 + 8x - 1
b ((5x^2 - 6x + 2) - (2x^2 - 4x + 1); 3x^2 - 2x + 1)c ((3x^2 + 5x + 4) + (x^2 + 2x - 5); 3x^2 + 7x - 1)d ((6x^2 - 3x - 8) - (3x^2 + 5x - 4); 3x^2 - 2x - 4)e ((x + 8)(x - 9); x^2 + x - 72)f ((x - 3)(x - 7); x^2 - 10x + 21)

Explanation:

Step1: Check Option a

Simplify \((3x^2 + 7x - 4)+(x^2 + x + 3)\):
Combine like terms: \(3x^2+x^2 = 4x^2\), \(7x+x = 8x\), \(-4 + 3=-1\). So it becomes \(4x^2 + 8x - 1\), which matches the second expression. So a is equivalent.

Step2: Check Option b

Simplify \((5x^2 - 6x + 2)-(2x^2 - 4x + 1)\):
Distribute the negative sign: \(5x^2 - 6x + 2-2x^2 + 4x - 1\). Combine like terms: \(5x^2-2x^2 = 3x^2\), \(-6x + 4x=-2x\), \(2 - 1 = 1\). So it becomes \(3x^2 - 2x + 1\), which matches the second expression. So b is equivalent.

Step3: Check Option c

Simplify \((3x^2 + 5x + 4)+(x^2 + 2x - 5)\):
Combine like terms: \(3x^2+x^2 = 4x^2\), \(5x+2x = 7x\), \(4-5=-1\). The result should be \(4x^2 + 7x - 1\), but the second expression is \(3x^2 + 7x - 1\). So c is not equivalent.

Step4: Check Option d

Simplify \((6x^2 - 3x - 8)-(3x^2 + 5x - 4)\):
Distribute the negative sign: \(6x^2 - 3x - 8-3x^2 - 5x + 4\). Combine like terms: \(6x^2-3x^2 = 3x^2\), \(-3x-5x=-8x\), \(-8 + 4=-4\). The result should be \(3x^2 - 8x - 4\), but the second expression is \(3x^2 - 2x - 4\). So d is not equivalent.

Step5: Check Option e

Simplify \((x + 8)(x - 9)\):
Use the FOIL method: \(x\cdot x=x^2\), \(x\cdot(-9)=-9x\), \(8\cdot x = 8x\), \(8\cdot(-9)=-72\). Combine like terms: \(x^2-9x + 8x-72=x^2 - x - 72\), but the second expression is \(x^2 + x - 72\). So e is not equivalent.

Step6: Check Option f

Simplify \((x - 3)(x - 7)\):
Use the FOIL method: \(x\cdot x=x^2\), \(x\cdot(-7)=-7x\), \(-3\cdot x=-3x\), \(-3\cdot(-7)=21\). Combine like terms: \(x^2-7x-3x + 21=x^2 - 10x + 21\), which matches the second expression. So f is equivalent.

Answer:

a, b, f