To expand the expression \((x - 1)(2x + 1)(x - 6)(x + 1)\), we can proceed step by step, expanding two factors at a time.
Step 1: Expand \((x - 1)(2x + 1)\)
\[ (x - 1)(2x + 1) = x(2x + 1) - 1(2x + 1) \] \[ = 2x^2 + x - 2x - 1 \] \[ = 2x^2 - x - 1 \]
Step 2: Expand \((x - 6)(x + 1)\)
\[ (x - 6)(x + 1) = x(x + 1) - 6(x + 1) \] \[ = x^2 + x - 6x - 6 \] \[ = x^2 - 5x - 6 \]
Step 3: Now multiply the two results: \((2x^2 - x - 1)(x^2 - 5x - 6)\)
We will use the distributive property (FOIL method) here:
\[ = 2x^2(x^2 - 5x - 6) - x(x^2 - 5x - 6) - 1(x^2 - 5x - 6) \]
Calculating each component:
- \(2x^2(x^2) = 2x^4\)
- \(2x^2(-5x) = -10x^3\)
- \(2x^2(-6) = -12x^2\)
Thus, we have:
\[ 2x^4 - 10x^3 - 12x^2 \]
Next:
- \(-x(x^2) = -x^3\)
- \(-x(-5x) = 5x^2\)
- \(-x(-6) = 6x\)
Thus, we have:
\[ -x^3 + 5x^2 + 6x \]
Lastly:
- \(-1(x^2) = -x^2\)
- \(-1(-5x) = 5x\)
- \(-1(-6) = 6\)
Thus, we have:
\[ -x^2 + 5x + 6 \]
Combining all parts:
\[ 2x^4 - 10x^3 - 12x^2 - x^3 + 5x^2 + 6x - x^2 + 5x + 6 \]
Now we combine like terms:
- \(2x^4\)
- \(-10x^3 - x^3 = -11x^3\)
- \(-12x^2 + 5x^2 - x^2 = -8x^2\)
- \(6x + 5x = 11x\)
- Constant term is \(6\)
Putting it all together, we get:
\[ \boxed{2x^4 - 11x^3 - 8x^2 + 11x + 6} \]
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