Expand the product of 5x2 + 6x and 9x2 - 8.

Step-by-step solution

1. 1

   Count the products before starting. Two terms times two terms gives $2\times 2=4$ products. Keeping track of the expected count is the simplest guard against dropping one.

2. 2

   Distribute the first term.

   $5x^2(9x^2-8)=45x^4-40x^2$

   Exponents add when multiplying: $x^2\cdot x^2=x^4$, not $x^{4}$ by squaring the coefficient.

3. 3

   Distribute the second term.

   $6x(9x^2-8)=54x^3-48x$

4. 4

   Add the two results and order by degree.

   $45x^4-40x^2+54x^3-48x=45x^4+54x^3-40x^2-48x$

   The four terms have degrees $4,3,2,1$ — all different — so there are no like terms to combine. Any "simplification" beyond reordering would be an error.

5. 5

   Check at $x=1$, and note the hidden common factor. The original is $(5+6)(9-8)=11$, and the expansion gives $45+54-40-48=11$ ✓. Every term contains an $x$, so the answer also factors as $x(5x+6)(9x^2-8)$ — a useful form if the next step is to solve for the roots.