Questions: Use the box method to distribute and simplify (-6+3x+5x^2)(-x+1).

Use the box method to distribute and simplify (-6+3x+5x^2)(-x+1).
Transcript text: Use the box method to distribute and simplify $\left(-6+3 x+5 x^{2}\right)(-x+1)$.
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Solution

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Solution Steps

Step 1: Distributing the Terms

We start with the expression \((-6 + 3x + 5x^{2})(-x + 1)\). Using the box method, we distribute each term in the first polynomial across each term in the second polynomial. This results in the following products:

  • \((-6)(-x) = 6x\)
  • \((-6)(1) = -6\)
  • \((3x)(-x) = -3x^{2}\)
  • \((3x)(1) = 3x\)
  • \((5x^{2})(-x) = -5x^{3}\)
  • \((5x^{2})(1) = 5x^{2}\)
Step 2: Combining Like Terms

Next, we combine all the products obtained from the distribution:

\[ 6x - 6 - 3x^{2} + 3x - 5x^{3} + 5x^{2} \]

Rearranging and combining like terms gives us:

\[ -5x^{3} + (5x^{2} - 3x^{2}) + (6x + 3x) - 6 \]

This simplifies to:

\[ -5x^{3} + 2x^{2} + 9x - 6 \]

Final Answer

The simplified expression for \((-6 + 3x + 5x^{2})(-x + 1)\) is

\[ \boxed{-5x^{3} + 2x^{2} + 9x - 6} \]

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