Expand and simplify: $$ \begin{array}{lll} ext { (1) } 2 x(3 x-4 y)-\left(7 x^{2}-2 x y ight) & ext { (2) } \quad(5 y+1)^{2}-(3 y+4)(2-3 y) \ ext { (3) }(2 x+y)^{2}-(3 x-2 y)^{2}+(x-4 y)(x+4 y) \ ext { (4) } x^{6}+\left(x^{3}-3 y ight)\left(x^{3}+3 y ight) & ext { (5) } \quad(3 a+b)(3 a-b)(2 a+5 b)\end{array} $$

Question

Expand and simplify: $$ \begin{array}{lll}	ext { (1) } 2 x(3 x-4 y)-\left(7 x^{2}-2 x yight) & 	ext { (2) } \quad(5 y+1)^{2}-(3 y+4)(2-3 y) \ 	ext { (3) }(2 x+y)^{2}-(3 x-2 y)^{2}+(x-4 y)(x+4 y) \ 	ext { (4) } x^{6}+\left(x^{3}-3 yight)\left(x^{3}+3 yight) & 	ext { (5) } \quad(3 a+b)(3 a-b)(2 a+5 b)\end{array} $$

UpStudy AI · Accepted Answer

Let's expand and simplify each expression step by step.

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**1. $$ 2x(3x - 4y) - (7x^{2} - 2xy) $$**

**Expansion:**
$$
2x(3x) + 2x(-4y) - 7x^{2} + 2xy = 6x^{2} - 8xy - 7x^{2} + 2xy
$$

**Combine like terms:**
$$
(6x^{2} - 7x^{2}) + (-8xy + 2xy) = -x^{2} - 6xy
$$

**Simplified Form:**
$$
- x^{2} - 6xy
$$

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**2. $$ (5y + 1)^{2} - (3y + 4)(2 - 3y) $$**

**Expansion:**
$$
(5y)^{2} + 2 \cdot 5y \cdot 1 + 1^{2} - [3y \cdot 2 + 3y \cdot (-3y) + 4 \cdot 2 + 4 \cdot (-3y)]
$$
$$
= 25y^{2} + 10y + 1 - (6y - 9y^{2} + 8 - 12y)
$$
$$
= 25y^{2} + 10y + 1 - 6y + 9y^{2} - 8 + 12y
$$

**Combine like terms:**
$$
(25y^{2} + 9y^{2}) + (10y - 6y + 12y) + (1 - 8) = 34y^{2} + 16y - 7
$$

**Simplified Form:**
$$
34y^{2} + 16y - 7
$$

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**3. $$ (2x + y)^{2} - (3x - 2y)^{2} + (x - 4y)(x + 4y) $$**

**Expansion:**
$$
(2x)^{2} + 2 \cdot 2x \cdot y + y^{2} - [(3x)^{2} + 2 \cdot 3x \cdot (-2y) + (-2y)^{2}] + (x^{2} - (4y)^{2})
$$
$$
= 4x^{2} + 4xy + y^{2} - (9x^{2} - 12xy + 4y^{2}) + x^{2} - 16y^{2}
$$
$$
= 4x^{2} + 4xy + y^{2} - 9x^{2} + 12xy - 4y^{2} + x^{2} - 16y^{2}
$$

**Combine like terms:**
$$
(4x^{2} - 9x^{2} + x^{2}) + (4xy + 12xy) + (y^{2} - 4y^{2} - 16y^{2}) = -4x^{2} + 16xy - 19y^{2}
$$

**Simplified Form:**
$$
-4x^{2} + 16xy - 19y^{2}
$$

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**4. $$ x^{6} + (x^{3} - 3y)(x^{3} + 3y) $$**

**Expansion using the difference of squares:**
$$
(x^{3} - 3y)(x^{3} + 3y) = x^{6} - (3y)^{2} = x^{6} - 9y^{2}
$$

**Combine with the existing $$ x^{6} $$:**
$$
x^{6} + x^{6} - 9y^{2} = 2x^{6} - 9y^{2}
$$

**Simplified Form:**
$$
2x^{6} - 9y^{2}
$$

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**5. $$ (3a + b)(3a - b)(2a + 5b) $$**

**First, expand $$ (3a + b)(3a - b) $$ using the difference of squares:**
$$
(3a + b)(3a - b) = 9a^{2} - b^{2}
$$

**Now, multiply by $$ (2a + 5b) $$:**
$$
(9a^{2} - b^{2})(2a) + (9a^{2} - b^{2})(5b)
$$
$$
= 18a^{3} - 2a b^{2} + 45a^{2}b - 5b^{3}
$$

**Arrange in standard form:**
$$
18a^{3} + 45a^{2}b - 2ab^{2} - 5b^{3}
$$

**Simplified Form:**
$$
18a^{3} + 45a^{2}b - 2ab^{2} - 5b^{3}
$$

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**Summary of Simplified Expressions:**
1. $$-x^{2} - 6xy$$
2. $$34y^{2} + 16y - 7$$
3. $$-4x^{2} + 16xy - 19y^{2}$$
4. $$2x^{6} - 9y^{2}$$
5. $$18a^{3} + 45a^{2}b - 2ab^{2} - 5b^{3}$$
1. $$-x^{2} - 6xy$$ 2. $$34y^{2} + 16y - 7$$ 3. $$-4x^{2} + 16xy - 19y^{2}$$ 4. $$2x^{6} - 9y^{2}$$ 5. $$18a^{3} + 45a^{2}b - 2ab^{2} - 5b^{3}$$

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