(4x + 1)(3x - 7) - (3x - 4)(2x - 8)

Steps to Solve

1. Expand the first expression: (4x + 1)(3x - 7)

Using the distributive property (also known as the FOIL method for binomials):

$$ (4x + 1)(3x - 7) = 4x \cdot 3x + 4x \cdot (-7) + 1 \cdot 3x + 1 \cdot (-7) $$

Calculating each part:

• $4x \cdot 3x = 12x^2$
• $4x \cdot (-7) = -28x$
• $1 \cdot 3x = 3x$
• $1 \cdot (-7) = -7$

So, combining these gives:

$$ 12x^2 - 28x + 3x - 7 = 12x^2 - 25x - 7 $$

2. Expand the second expression: (3x - 4)(2x - 8)

Again using the distributive property:

$$ (3x - 4)(2x - 8) = 3x \cdot 2x + 3x \cdot (-8) + (-4) \cdot 2x + (-4) \cdot (-8) $$

Calculating each part:

• $3x \cdot 2x = 6x^2$
• $3x \cdot (-8) = -24x$
• $-4 \cdot 2x = -8x$
• $-4 \cdot (-8) = 32$

So, combining these gives:

$$ 6x^2 - 24x - 8x + 32 = 6x^2 - 32x + 32 $$

3. Subtract the second expression from the first

Now we combine the two expanded expressions from the previous steps:

$$ (12x^2 - 25x - 7) - (6x^2 - 32x + 32) $$

Distributing the negative sign:

$$ 12x^2 - 25x - 7 - 6x^2 + 32x - 32 $$

4. Combine like terms

Combine the terms:

• For $x^2$ terms: $12x^2 - 6x^2 = 6x^2$
• For $x$ terms: $-25x + 32x = 7x$
• For constant terms: $-7 - 32 = -39$

So, putting them all together gives:

$$ 6x^2 + 7x - 39 $$