Expand and simplify: $$ \begin{array}{ll} ext { (a) } 2 x(3 x-4 y)-\left(7 x^{2}-2 x y ight) & ext { (b) } \ ext { (c) } r^{10} & (5 y+1)^{2}-(3 y+4)(2-3 y) \ ext { (d) }(8 m-y)^{2}-(3 x-2 y)^{2}+(x-4 y)(x+4 y) \ ext { (c) } x^{6}+\left(x^{3}-3 y ight)\left(x^{3}+3 y ight) & ext { (f) }\end{array} $$

Question

Expand and simplify: $$ \begin{array}{ll}	ext { (a) } 2 x(3 x-4 y)-\left(7 x^{2}-2 x yight) & 	ext { (b) } \ 	ext { (c) } r^{10} & (5 y+1)^{2}-(3 y+4)(2-3 y) \ 	ext { (d) }(8 m-y)^{2}-(3 x-2 y)^{2}+(x-4 y)(x+4 y) \ 	ext { (c) } x^{6}+\left(x^{3}-3 yight)\left(x^{3}+3 yight) & 	ext { (f) }\end{array} $$

UpStudy AI · Accepted Answer

Sure, let's expand and simplify each of the given expressions step by step.

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### **(a) $$2x(3x - 4y) - (7x^2 - 2xy)$$**

**Step 1:** Distribute the terms.
$$
2x \cdot 3x = 6x^2 \
2x \cdot (-4y) = -8xy
$$

**Step 2:** Distribute the negative sign in the second part.
$$
-(7x^2 - 2xy) = -7x^2 + 2xy
$$

**Step 3:** Combine like terms.
$$
6x^2 - 8xy - 7x^2 + 2xy = (6x^2 - 7x^2) + (-8xy + 2xy) = -x^2 - 6xy
$$

**Final Answer:**
$$
-\,x^2 - 6xy
$$

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### **(b) *(No expression provided.)***

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### **(c) $$(5y + 1)^2 - (3y + 4)(2 - 3y)$$**

**Step 1:** Expand $$(5y + 1)^2$$.
$$
(5y)^2 + 2 \cdot 5y \cdot 1 + 1^2 = 25y^2 + 10y + 1
$$

**Step 2:** Expand $$(3y + 4)(2 - 3y)$$.
$$
3y \cdot 2 + 3y \cdot (-3y) + 4 \cdot 2 + 4 \cdot (-3y) = 6y - 9y^2 + 8 - 12y = -9y^2 - 6y + 8
$$

**Step 3:** Subtract the second expansion from the first.
$$
25y^2 + 10y + 1 - (-9y^2 - 6y + 8) = 25y^2 + 10y + 1 + 9y^2 + 6y - 8 = 34y^2 + 16y - 7
$$

**Final Answer:**
$$
34y^2 + 16y - 7
$$

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### **(d) $$(8m - y)^2 - (3x - 2y)^2 + (x - 4y)(x + 4y)$$**

**Step 1:** Expand each squared binomial and the product.
$$
(8m - y)^2 = 64m^2 - 16my + y^2 \
(3x - 2y)^2 = 9x^2 - 12xy + 4y^2 \
(x - 4y)(x + 4y) = x^2 - 16y^2
$$

**Step 2:** Combine all the terms.
$$
64m^2 - 16my + y^2 - (9x^2 - 12xy + 4y^2) + x^2 - 16y^2
$$

**Step 3:** Distribute the negative sign and combine like terms.
$$
64m^2 - 16my + y^2 - 9x^2 + 12xy - 4y^2 + x^2 - 16y^2 = 64m^2 - 8x^2 - 16my + 12xy - 19y^2
$$

**Final Answer:**
$$
64m^2 - 8x^2 - 16my + 12xy - 19y^2
$$

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### **(e) $$x^6 + (x^3 - 3y)(x^3 + 3y)$$**

**Step 1:** Recognize the difference of squares in the second term.
$$
(x^3 - 3y)(x^3 + 3y) = x^6 - (3y)^2 = x^6 - 9y^2
$$

**Step 2:** Combine with the first term.
$$
x^6 + (x^6 - 9y^2) = 2x^6 - 9y^2
$$

**Final Answer:**
$$
2x^6 - 9y^2
$$

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### **(f) *(No expression provided.)***

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If you have any additional parts or need further clarification on any of these steps, feel free to ask!
(a) $$ -x^2 - 6xy $$ (b) *(No expression provided.)* (c) $$ 34y^2 + 16y - 7 $$ (d) $$ 64m^2 - 8x^2 - 16my + 12xy - 19y^2 $$ (e) $$ 2x^6 - 9y^2 $$ (f) *(No expression provided.)*

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