Explanation

We want to simplify the expression:

\(\binom{2x + 1}{3} - \binom{2x - 1}{3} - 2\binom{x}{2}\)

Write the Binomial Coefficients

1. First term:
\(\binom{2x + 1}{3} = \frac{(2x + 1)(2x)(2x - 1)}{6}\) from the fact that \(\binom{2x + 1}{3}\) = \(\frac{(2x + 1)!}{3!(3 - (2x + 1)!}\) = \(\frac{(2x + 1)(2x)(2x - 2)!}{6(2x - 2)!}\)

After cancelling, we have  \(\frac{(2x + 1)(2x)(2x - 1)}{6}\).

2. Second term:
\(\binom{2x - 1}{3} = \frac{(2x - 1)(2x - 2)(2x - 3)}{6}\)

3. Third term:
\(2\binom{x}{2} = 2 \cdot \frac{x(x - 1)}{2} = x(x - 1)\)

Combine the First Two Terms

Combining the first two terms gives us: \(\frac{(2x + 1)(2x)(2x - 1) - (2x - 1)(2x - 2)(2x - 3)}{6}\)

 Expand the Numerators

1. Expanding \((2x + 1)(2x)(2x - 1)\): \((2x)(2x - 1) = 4x^2 - 2x\)
\((2x + 1)(4x^2 - 2x) = 8x^3 - 4x^2 + 4x^2 - 2x = 8x^3 - 2x\)

 Expanding \((2x - 1)(2x - 2)(2x - 3)\):
First, calculate:
\((2x - 2)(2x - 3) = 4x^2 - 10x + 6\), Then multiply by \((2x - 1)\):
\((2x - 1)(4x^2 - 10x + 6) = 8x^3 - 20x^2 + 12x - 4x^2 + 10x - 6 = 8x^3 - 24x^2 + 22x - 6\)

Substitute Back into the Expression, Now we have:
\(\frac{(8x^3 - 2x) - (8x^3 - 24x^2 + 22x - 6)}{6}\)

Simplify the Numerator; Distributing the negative:
\(= \frac{8x^3 - 2x - 8x^3 + 24x^2 - 22x + 6}{6}\)
Combining like terms: \(= \frac{24x^2 - 24x + 6}{6}\)

 Include \(x(x - 1)\)

Now subtract \(x(x - 1)\):
\(= \frac{24x^2 - 24x + 6}{6} - x(x - 1)\)
Convert \(x(x - 1)\) into sixths:
\(= \frac{24x^2 - 24x + 6 - 6(x^2 - x)}{6}\)
\(= \frac{24x^2 - 24x + 6 - 6x^2 + 6x}{6}\)

Now, combine the terms in the numerator:
\(= \frac{(24x^2 - 6x^2) + (-24x + 6x) + 6}{6}\)
\(= \frac{18x^2 - 18x + 6}{6}\)

Now divide each term by 6:
\(= 3x^2 - 3x + 1\).

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