To determine which expression is equivalent to the polynomial [tex]\(6x^2 + 10x - 56\)[/tex], we will evaluate each given option.
### Option 1: [tex]\(4(x-4) - 2(3x^2 + 3x - 20)\)[/tex]
First, expand [tex]\(4(x-4)\)[/tex]:
[tex]\[ 4(x-4) = 4x - 16 \][/tex]
Next, expand [tex]\(-2(3x^2 + 3x - 20)\)[/tex]:
[tex]\[ -2(3x^2 + 3x - 20) = -2 \cdot 3x^2 - 2 \cdot 3x - 2 \cdot (-20) = -6x^2 - 6x + 40 \][/tex]
Now, combine both results:
[tex]\[ 4x - 16 - 6x^2 - 6x + 40 \][/tex]
Combine like terms:
[tex]\[ -6x^2 - 2x + 24 \][/tex]
This is not equivalent to [tex]\(6x^2 + 10x - 56\)[/tex].
### Option 2: [tex]\((3x^2 + 5x - 28) + (3x^2 - 5x - 28)\)[/tex]
First, add the polynomials term by term:
[tex]\[ (3x^2 + 5x - 28) + (3x^2 - 5x - 28) \][/tex]
Combine like terms:
[tex]\[ 3x^2 + 3x^2 + 5x - 5x - 28 - 28 \][/tex]
[tex]\[ 6x^2 - 56 \][/tex]
This is not equivalent to [tex]\(6x^2 + 10x - 56\)[/tex].
### Option 3: [tex]\(4(x-4) + 2(3x^2 + 3x - 20)\)[/tex]
First, expand [tex]\(4(x-4)\)[/tex]:
[tex]\[ 4(x-4) = 4x - 16 \][/tex]
Next, expand [tex]\(2(3x^2 + 3x - 20)\)[/tex]:
[tex]\[ 2(3x^2 + 3x - 20) = 2 \cdot 3x^2 + 2 \cdot 3x + 2 \cdot (-20) = 6x^2 + 6x - 40 \][/tex]
Now, combine both results:
[tex]\[ 4x - 16 + 6x^2 + 6x - 40 \][/tex]
Combine like terms:
[tex]\[ 6x^2 + 10x - 56 \][/tex]
This matches the original polynomial [tex]\(6x^2 + 10x - 56\)[/tex].
### Option 4: [tex]\((3x^2 + 5x - 28) - (3x^2 + 5x - 28)\)[/tex]
First, subtract the polynomials term by term:
[tex]\[ (3x^2 + 5x - 28) - (3x^2 + 5x - 28) \][/tex]
Combine like terms:
[tex]\[ 3x^2 - 3x^2 + 5x - 5x - 28 + 28 \][/tex]
[tex]\[ 0 \][/tex]
This is not equivalent to [tex]\(6x^2 + 10x - 56\)[/tex].
Therefore, the correct expression equivalent to [tex]\(6x^2 + 10x - 56\)[/tex] is:
[tex]\[ \boxed{4(x-4) + 2(3x^2 + 3x - 20)} \][/tex]
