Expand and simplify (7x^2 + 7x - 5)(5x^2 - 4x + 2).

Step-by-step solution

1. 1

   Plan the bookkeeping before multiplying. Two trinomials produce $3\times3=9$ partial products. Distributing one left-hand term at a time (rather than FOIL, which only covers binomials) keeps all nine accounted for.

2. 2

   Distribute the leading term $7x^{2}$.

   $7x^{2}\left(5x^{2}-4x+2\right)=35x^{4}-28x^{3}+14x^{2}$

3. 3

   Distribute the middle term $7x$.

   $7x\left(5x^{2}-4x+2\right)=35x^{3}-28x^{2}+14x$

4. 4

   Distribute the constant $-5$, carrying its sign through every term.

   $-5\left(5x^{2}-4x+2\right)=-25x^{2}+20x-10$

5. 5

   Collect like powers. Group the nine terms by degree:

   $x^{4}:\ 35x^{4}$
   $x^{3}:\ -28x^{3}+35x^{3}=7x^{3}$
   $x^{2}:\ 14x^{2}-28x^{2}-25x^{2}=-39x^{2}$
   $x^{1}:\ 14x+20x=34x$
   $x^{0}:\ -10$

6. 6

   Check the expansion numerically. Evaluating the factored form and the expanded form at every integer $x$ from $-6$ to $6$ gives identical values at all $13$ points — more than the $5$ needed to determine a quartic, so the coefficients are correct. (Quick single check at $x=1$: $(7+7-5)(5-4+2)=9\cdot3=27$ and $35+7-39+34-10=27$.)