Thank you for your reply, but I cannot read you post after -(2a - 3b)x + 6.

Secondly, and I am probably wrong as you are more knowledgeable, but when you multiply f(x)g(x) I get: 6X^4 - 2bX^3 -**4x^2 (you got 2x^2)** + 3aX^3 - **abx^2** - 2ax - 9x^2 + 3bx + 6 **The term in blue is incorrect. See below**

So when multiplying by 2x^2, you get 6X^4 - 2bX^3 - 4x^2 (2x^2 * -2) ; it is when multiplying g(x) by 2x^2 that we get different answers. If I am right I assume simplifying the answer will yield different results.

Also, do you use a programme or software to write in that easy to read bold font, it beats Microsoft word!

Don't worry about the scripting language: it is fussy.

\(\displaystyle f(x) = 2x^2 + ax - 3\ and\ g(x) = 3x^2 - bx - 2 \implies f(x) * g(x) = (2x^2 + ax - 3)(3x^2 - bx - 2) =\)

\(\displaystyle f(x) = 2x^2(3x^2 - bx - 2) + ax(3x^2 - bx - 2) - 3(3x^2 - bx - 2) =\)

\(\displaystyle 6x^4 - 2bx^3 - 4x^2 + 3ax^3 - abx^2 - 2ax - 9x^2 + 3bx + 6 =\)

\(\displaystyle 6x^4 + (3a - 2b)x^3 - (13 + ab)x^2 + (3b - 2a)x + 6.\)

You did not expand the function correctly. Notice the coefficient of the x squared term.

By the terms of the problem

\(\displaystyle 3a - 2b = 6 \implies a = 2 + \dfrac{2b}{3}.\)

Also \(\displaystyle 3b - 2a = 1 \implies 3b - 2\left(2 + \dfrac{2b}{3}\right) = 1 \implies 3b - 4 - \dfrac{4b}{3} = 1 \implies\)

\(\displaystyle \dfrac{9b}{3} - \dfrac{4b}{3} = 4 + 1 = 5 \implies \dfrac{5b}{3} = 5 \implies b = \dfrac{3 * 5}{5} = 3 \implies\)

\(\displaystyle a = 2 + \dfrac{2 * 3}{3} = 4 \implies\)

\(\displaystyle 13 + ab = 13 + 4 * 3 = 25.\)

Now let's check.

\(\displaystyle (2x^2 + 4x - 3)(3x^2 - 3x - 2) = 2x^2(3x^2 - 3x - 2) + 4x(3x^2 - 3x - 2) - 3(3x^2 - 3x - 2) =\)

\(\displaystyle 6x^4 - 6x^3 - 4x^2 + 12x^3 - 12x^2 - 8x - 9x^2 + 9x + 6 = 6x^4 + 6x^3 - 25x^2 + x + 6.\)

You had the right idea. You made a careless mistake. To be honest, so did I the first time through. But checking my answer allowed me to find my mistake.