Multiply by using the Distributive Property. 1. $$ (3 x+1)\left(x^{3}+4 x^{2}-7\right) $$ $$ (2 x-5)\left(4 x^{2}-3 x+1\right) $$ 2.

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### Problem 1

Multiply $$ (3x+1)(x^3+4x^2-7) $$ using the Distributive Property.

1. Distribute $$ 3x $$ over the second polynomial:
   - $$3x \cdot x^3 = 3x^4$$
   - $$3x \cdot 4x^2 = 12x^3$$
   - $$3x \cdot (-7) = -21x$$

2. Distribute $$ 1 $$ over the second polynomial:
   - $$1 \cdot x^3 = x^3$$
   - $$1 \cdot 4x^2 = 4x^2$$
   - $$1 \cdot (-7) = -7$$

3. Combine like terms:
   - $$3x^4$$ (no like term)
   - $$12x^3 + x^3 = 13x^3$$
   - $$4x^2$$ (no like term)
   - $$-21x$$ (no like term)
   - $$-7$$ (constant)

The expanded form is:
$$
3x^4 + 13x^3 + 4x^2 -21x -7
$$

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### Problem 2

Multiply $$ (2x-5)(4x^2-3x+1) $$ using the Distributive Property.

1. Distribute $$ 2x $$ over the second polynomial:
   - $$2x \cdot 4x^2 = 8x^3$$
   - $$2x \cdot (-3x) = -6x^2$$
   - $$2x \cdot 1 = 2x$$

2. Distribute $$-5$$ over the second polynomial:
   - $$-5 \cdot 4x^2 = -20x^2$$
   - $$-5 \cdot (-3x) = 15x$$
   - $$-5 \cdot 1 = -5$$

3. Combine like terms:
   - $$8x^3$$ (no like term)
   - $$-6x^2 -20x^2 = -26x^2$$
   - $$2x + 15x = 17x$$
   - $$-5$$ (constant)

The expanded form is:
$$
8x^3 -26x^2 +17x -5
$$
$$ 3x^4 + 13x^3 + 4x^2 -21x -7 $$ $$ 8x^3 -26x^2 +17x -5 $$

Multiply by using the Distributive Property.

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