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Identify all possible factorizations of the binomial.

8x^2+12x

Responses

6x(x+2)
6 x open paren x plus 2 close paren

3x(2x+4)
3 x open paren 2 x plus 4 close paren

2x2(4x+6)
2 x squared open paren 4 x plus 6 close paren

4(2x+3)
4 times open paren 2 x plus 3 close paren

4x(2x+3)
4 x open paren 2 x plus 3 close paren

2x(4x+6)

1 answer

To factor the expression \(8x^2 + 12x\), we can start by identifying the greatest common factor (GCF) of the terms in the binomial.

  1. Find the GCF: The coefficients are 8 and 12. The GCF of 8 and 12 is 4. Additionally, both terms contain \(x\). The GCF of the entire expression \(8x^2 + 12x\) is \(4x\).

  2. Factor out the GCF: We can factor \(4x\) out of each term:

    \[ 8x^2 + 12x = 4x(2x + 3) \]

From this factorization, we can identify the following possible factorizations for \(8x^2 + 12x\):

  1. Correct Factorization:
    • \(4x(2x + 3)\)

We can express \(8x^2 + 12x\) in terms of its factors directly, and we can check the other options given:

  • \(6x(x + 2)\) does not reproduce the original expression when expanded.
  • \(3x(2x + 4)\) does not reproduce the original expression.
  • \(2x^2(4x + 6)\) is not appropriate since \(2x^2\) cannot be factored out correctly.
  • \(4(2x + 3)\) is missing the \(x\) factor.
  • \(4x(2x + 3)\) is valid and matches our derived factorization.
  • \(2x(4x + 6)\) is also incorrect for the same reason previously stated.

Thus, the only correct factorization for \(8x^2 + 12x\) is:

\[ \boxed{4x(2x + 3)} \]