Solve for [tex] x [/tex]:

[tex]\[ 3x = 6x - 2 \][/tex]

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Format the following question or task so that it is easier to read. Fix any grammar or spelling errors. Remove phrases that are not part of the question. Do not remove or change LaTeX formatting. Do not change or remove [tex] [/tex] tags. If the question is nonsense, rewrite it so that it makes sense.

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[tex]\[\begin{array}{l} -3A(x) + 5B(x) = \\ \left(7 - 8x^3 - \frac{3}{2}x + \frac{5}{2}x\right) \left(4x^3 - 3x + 5\right) \end{array} \][/tex]



Answer :

Sure, let's solve the given polynomial multiplication step-by-step.

Given the polynomial equation:

[tex]\[ \left(7 - 8x^3 - \frac{3}{2}x + \frac{5}{2}x \right)\left(4x^3 - 3x + 5\right) \][/tex]

First, we simplify the terms inside each of the polynomial expressions.

For polynomial \(A(x)\):
[tex]\[ A(x) = 7 - 8x^3 - \frac{3}{2}x + \frac{5}{2}x \][/tex]
Combining like terms:
[tex]\[ A(x) = 7 - 8x^3 + x \][/tex]

Next, for polynomial \(B(x)\):
[tex]\[ B(x) = 4x^3 - 3x + 5 \][/tex]

Now, we need to multiply these polynomials together. The multiplication of two polynomials is performed by distributing every term in \(A(x)\) to every term in \(B(x)\).

[tex]\[ (7 - 8x^3 + x)(4x^3 - 3x + 5) \][/tex]

Distribute each term of \(7\):
[tex]\[ 7 \cdot 4x^3 = 28x^3 \][/tex]
[tex]\[ 7 \cdot (-3x) = -21x \][/tex]
[tex]\[ 7 \cdot 5 = 35 \][/tex]

Distribute each term of \(-8x^3\):
[tex]\[ -8x^3 \cdot 4x^3 = -32x^6 \][/tex]
[tex]\[ -8x^3 \cdot (-3x) = 24x^4 \][/tex]
[tex]\[ -8x^3 \cdot 5 = -40x^3 \][/tex]

Distribute each term of \(x\):
[tex]\[ x \cdot 4x^3 = 4x^4 \][/tex]
[tex]\[ x \cdot (-3x) = -3x^2 \][/tex]
[tex]\[ x \cdot 5 = 5x \][/tex]

Now we combine all the results:
[tex]\[ -32x^6 + 24x^4 + 4x^4 + 28x^3 - 40x^3 -3x^2 - 21x + 5x + 35 \][/tex]

Next, combine like terms:
[tex]\[ -32x^6 + (24x^4 + 4x^4) + (28x^3 - 40x^3) - 3x^2 + (-21x + 5x) + 35 \][/tex]

Simplify the coefficients:
[tex]\[ -32x^6 + 28x^4 - 12x^3 - 3x^2 - 16x + 35 \][/tex]

Therefore, the final simplified polynomial is:
[tex]\[ -32x^6 + 28.0x^4 - 12x^3 - 3.0x^2 - 16.0x + 35 \][/tex]