Expand and simplify (10x^3 + 3x + 8)(4x^3 - 8x - 9).

Step-by-step solution

1. 1

   Plan the expansion. Each factor has three terms, so the distributive law produces $3\times 3=9$ products. Note that neither factor has an $x^{2}$ term — that missing degree is exactly where sign and bookkeeping errors creep in, so track degrees explicitly.

2. 2

   Distribute the leading term $10x^{3}$.

   $10x^{3}\left(4x^{3}-8x-9\right)=40x^{6}-80x^{4}-90x^{3}$

3. 3

   Distribute the middle term $3x$.

   $3x\left(4x^{3}-8x-9\right)=12x^{4}-24x^{2}-27x$

4. 4

   Distribute the constant term $8$.

   $8\left(4x^{3}-8x-9\right)=32x^{3}-64x-72$

5. 5

   Collect the nine terms by degree. Line them up degree by degree:

   $\begin{aligned}x^{6}&:\ 40x^{6}\\ x^{5}&:\ \text{none}\\ x^{4}&:\ -80x^{4}+12x^{4}=-68x^{4}\\ x^{3}&:\ -90x^{3}+32x^{3}=-58x^{3}\\ x^{2}&:\ -24x^{2}\\ x&:\ -27x-64x=-91x\\ \text{const}&:\ -72\end{aligned}$

6. 6

   Write the result and check the ends. The expansion is

   $40x^{6}-68x^{4}-58x^{3}-24x^{2}-91x-72$

   Two quick checks: the leading coefficient must be $10\times 4=40$ ✓ and the constant term must be $8\times(-9)=-72$ ✓. A third check is to evaluate both forms at $x=1$: the factors give $(10+3+8)(4-8-9)=21\times(-13)=-273$, and the expansion gives $40-68-58-24-91-72=-273$ ✓