Question

expand the expression to a polynomial in standard form:
$(x - 1)(2x - 3)(x - 4)(x + 1)$

Explanation:

Step1: Group and multiply pairs

First, group \((x - 1)\) and \((x + 1)\), also \((2x - 3)\) and \((x - 4)\).
For \((x - 1)(x + 1)\), use the difference of squares formula \((a - b)(a + b)=a^2 - b^2\), where \(a = x\), \(b = 1\). So \((x - 1)(x + 1)=x^2 - 1\).
For \((2x - 3)(x - 4)\), use the distributive property (FOIL method):

$$
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$$

Step2: Multiply the two resulting polynomials

Now we have \((x^2 - 1)(2x^2-11x + 12)\). Use the distributive property (multiply each term in the first polynomial by each term in the second polynomial):

$$
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$$

Step3: Combine like terms

Combine the \(x^2\) terms: \(12x^2-2x^2 = 10x^2\).
So the polynomial becomes \(2x^4-11x^3 + 10x^2 + 11x - 12\).

Answer:

\(2x^4-11x^3 + 10x^2 + 11x - 12\)