Question

multiply each of the following below and simplify if necessary.

1. $5x^2(3x^3 + 2x^2 - 5x - 9)$
2. $2xy^2(x^2 - 5xy + y^5)$
3. $(2x + 3)(5x - 7)$

$2x+3$

1. $(x^2 - 3y^2)^2$

$(x^2-3y^2)(x^2-3y^2)$
$x^2(x^2-3y^2)-3y^2(x^2-3y^2)$

Explanation:

Step1: Distribute $5x^2$ to each term

$5x^2 \cdot 3x^3 + 5x^2 \cdot 2x^2 + 5x^2 \cdot (-5x) + 5x^2 \cdot (-9)$

Step2: Simplify each product

$15x^{5} + 10x^{4} - 25x^{3} - 45x^{2}$

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Step1: Distribute $2xy^2$ to each term

$2xy^2 \cdot x^2 + 2xy^2 \cdot (-5xy) + 2xy^2 \cdot y^5$

Step2: Simplify each product

$2x^{3}y^{2} - 10x^{2}y^{3} + 2xy^{7}$

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Step1: Apply FOIL method

$2x \cdot 5x + 2x \cdot (-7) + 3 \cdot 5x + 3 \cdot (-7)$

Step2: Simplify each product

$10x^2 - 14x + 15x - 21$

Step3: Combine like terms

$10x^2 + x - 21$

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Step1: Expand using distribution

$x^2(x^2 - 3y^2) - 3y^2(x^2 - 3y^2)$

Step2: Distribute each term

$x^4 - 3x^2y^2 - 3x^2y^2 + 9y^4$

Step3: Combine like terms

$x^4 - 6x^2y^2 + 9y^4$

Answer:

1. $15x^{5} + 10x^{4} - 25x^{3} - 45x^{2}$
2. $2x^{3}y^{2} - 10x^{2}y^{3} + 2xy^{7}$
3. $10x^2 + x - 21$
4. $x^4 - 6x^2y^2 + 9y^4$