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Here are the simplified results for each expression:
1. \( (2x + 3)(4x^{2} - 6x + 9) = 8x^{3} + 27 \)
2. \( 2(x - y)^{2} + (x + 2y)(x - 2y) = 3x^{2} - 4xy - 2y^{2} \)
3. \( \left(a^{2} - 4\right)^{2} - \left(a^{2} + 1\right)\left(a^{2} - 1\right) + \left(2a^{2}\right)^{2} = 4a^{4} - 8a^{2} + 17 \)
4. \( -3x(3 - x)(2x^{2} - x - 5) = -21x^{3} - 6x^{2} + 45x + 6x^{4} \)
5. \( \left(\frac{1}{2}x - 3\right)\left(\frac{1}{4}x^{2} + \frac{3}{2}x + 9\right) = \frac{1}{8}x^{3} - 27 \)
Solución
Simplify the expression by following steps:
- step0: Calculate:
\(\left(\frac{1}{2}x-3\right)\left(\frac{1}{4}x^{2}+\frac{3}{2}x+9\right)\)
- step1: Multiply the expression:
\(2^{-3}\left(x^{3}-216\right)\)
- step2: Rearrange the terms:
\(2^{-3}x^{3}-27\)
- step3: Evaluate the power:
\(\frac{1}{8}x^{3}-27\)
Expand the expression \( 2(x-y)^{2}+(x+2 y)(x-2 y) \)
Simplify the expression by following steps:
- step0: Add the terms:
\(2\left(x-y\right)^{2}+\left(x+2y\right)\left(x-2y\right)\)
- step1: Expand the expression:
\(2x^{2}-4xy+2y^{2}+\left(x+2y\right)\left(x-2y\right)\)
- step2: Expand the expression:
\(2x^{2}-4xy+2y^{2}+x^{2}-4y^{2}\)
- step3: Add the terms:
\(3x^{2}-4xy-2y^{2}\)
Expand the expression \( (2 x+3)(4 x^{2}-6 x+9) \)
Simplify the expression by following steps:
- step0: Simplify the product:
\(\left(2x+3\right)\left(4x^{2}-6x+9\right)\)
- step1: Simplify the product:
\(8x^{3}+27\)
Expand the expression \( (a^{2}-4)^{2}-(a^{2}+1)(a^{2}-1)+(2 a^{2})^{2} \)
Simplify the expression by following steps:
- step0: Simplify:
\(\left(a^{2}-4\right)^{2}-\left(a^{2}+1\right)\left(a^{2}-1\right)+\left(2a^{2}\right)^{2}\)
- step1: Multiply the terms:
\(\left(a^{2}-4\right)^{2}+\left(-a^{2}-1\right)\left(a^{2}-1\right)+\left(2a^{2}\right)^{2}\)
- step2: Expand the expression:
\(a^{4}-8a^{2}+16+\left(-a^{2}-1\right)\left(a^{2}-1\right)+\left(2a^{2}\right)^{2}\)
- step3: Expand the expression:
\(a^{4}-8a^{2}+16+1-a^{4}+\left(2a^{2}\right)^{2}\)
- step4: Rewrite the expression:
\(a^{4}-8a^{2}+16+1-a^{4}+4a^{4}\)
- step5: Calculate:
\(4a^{4}-8a^{2}+17\)
Expand the expression \( -3 x(3-x)(2 x^{2}-x-5) \)
Simplify the expression by following steps:
- step0: Calculate:
\(-3x\left(3-x\right)\left(2x^{2}-x-5\right)\)
- step1: Multiply the terms:
\(\left(-9x+3x^{2}\right)\left(2x^{2}-x-5\right)\)
- step2: Apply the distributive property:
\(-9x\times 2x^{2}-\left(-9x\times x\right)-\left(-9x\times 5\right)+3x^{2}\times 2x^{2}-3x^{2}\times x-3x^{2}\times 5\)
- step3: Multiply the terms:
\(-18x^{3}-\left(-9x^{2}\right)-\left(-45x\right)+6x^{4}-3x^{3}-15x^{2}\)
- step4: Remove the parentheses:
\(-18x^{3}+9x^{2}+45x+6x^{4}-3x^{3}-15x^{2}\)
- step5: Subtract the terms:
\(-21x^{3}-6x^{2}+45x+6x^{4}\)
Here are the simplified results for each expression:
### 2.1) Simplify
#### 2.1.1)
\[
(2x + 3)(4x^{2} - 6x + 9) = 8x^{3} + 27
\]
#### 2.1.2)
\[
2(x - y)^{2} + (x + 2y)(x - 2y) = 3x^{2} - 4xy - 2y^{2}
\]
### 2.2)
\[
\left(a^{2} - 4\right)^{2} - \left(a^{2} + 1\right)\left(a^{2} - 1\right) + \left(2a^{2}\right)^{2} = 4a^{4} - 8a^{2} + 17
\]
### 2.3)
\[
-3x(3 - x)(2x^{2} - x - 5) = -21x^{3} - 6x^{2} + 45x + 6x^{4}
\]
### 2.4)
\[
\left(\frac{1}{2}x - 3\right)\left(\frac{1}{4}x^{2} + \frac{3}{2}x + 9\right) = \frac{1}{8}x^{3} - 27
\]
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