Question

How do you multiply `(x^2 + 3x + 1)(x^2 + x + 2)`?

Answer 1

`x^4 + 4x^3 + 6x^2 + 7x + 2`

Explanation:

We can think about this as distribution. We normally use FOIL for multiplying two binomials, but here we have trinomials, so that trick won't work.

However, we can think about what addition and multiplication mean together. We see this with FOIL:
`(a+b)(c+d) = a * c + a * d + b * c + b * d`
But we can also see that this can be factored a little:
`= a* (c + d) + b * (c+d)`
This is just showing distributive property of multiplication!

We can break down the above in a similar way to simplify the expression:

`(x^2+3x+1)(x^2+x+2)`
`= x^2 * (x^2+ x + 2) + 3x * (x^2+ x + 2) + 1 * (x^2+ x + 2)`
And now we distribute and combine like terms
`= x^4 + x^3 + 2x^2 + 3x^3 + 3x^2 + 6x + x^2 + x + 2`
`= x^4 + 4x^3 + 6x^2 + 7x + 2`

That strategy (of distributing each term individually) works for any polynomial multiplication.