To determine which expression is equivalent to [tex]\((7x - 3)(2x - 4)\)[/tex], we can use the distributive property (also known as the FOIL method) to expand the product of the binomials step-by-step.
Let's expand the expression [tex]\((7x - 3)(2x - 4)\)[/tex] by multiplying each term in the first binomial by each term in the second binomial:
1. First terms: Multiply the first term of each binomial:
[tex]\[(7x) \cdot (2x) = 14x^2\][/tex]
2. Outer terms: Multiply the outer terms:
[tex]\[(7x) \cdot (-4) = -28x\][/tex]
3. Inner terms: Multiply the inner terms:
[tex]\[(-3) \cdot (2x) = -6x\][/tex]
4. Last terms: Multiply the last term of each binomial:
[tex]\[(-3) \cdot (-4) = 12\][/tex]
So, combining all these steps, we have the expanded expression:
[tex]\[14x^2 - 28x - 6x + 12\][/tex]
Simplifying by combining the like terms [tex]\(-28x\)[/tex] and [tex]\(-6x\)[/tex], we get:
[tex]\[14x^2 - 34x + 12\][/tex]
Now, we compare this to the given options to find the one that matches our expanded expression.
Reviewing the given options:
1. [tex]\((7 x)(2 x)+(7 x)(4)+(-3)(2 x)+(-3)(-4)\)[/tex]
2. [tex]\((7 x)(2 x)+(7 x)(-4)+(3)(2 x)+(3)(-4)\)[/tex]
3. [tex]\((7 x)(2 x)+(7 x)(-4)+(-3)(2 x)+(-3)(-4)\)[/tex]
4. [tex]\((7 x)(2 x)+(-3)(-4)\)[/tex]
From these, option 3 correctly shows the steps taken during the expansion process:
[tex]\[(7x)(2x) + (7x)(-4) + (-3)(2x) + (-3)(-4)\][/tex]
Thus, the correct choice equivalent to [tex]\((7x - 3)(2x - 4)\)[/tex] is:
[tex]\[(7 x)(2 x)+(7 x)(-4)+(-3)(2 x)+(-3)(-4)\][/tex]
