Let's first look at the possible combinations of terms that could give us an "x3" term. There are 2 cases
1. Constant multiplied by an x^3 term(for example, x3 * 6)
2. A first power of x multiplied by a second power
Let's start with the first case, with a constant multiplied by an x3 term. We see that we are given two equations:
\(24x^4 + 6x^3 + 4x^2-7x - 5\) and \(6x^3 + 3x^2 + 3x + 4\). If we're looking at a constant multiplied by an x3 term, there are two such "pairs" when we multiply these two polynomials:
\(6x^3*4 +6x^3*(-5) = 24x^3-30x^3 = -6x^3\)(Do you see how I got this? When you multiply the two polynomials, you can think of it as "cross multiplication" and take the desired pairs. That gives us our x3 term for our first case. Let's then move on to the second case:
A first power of x multiplied by a second power, or vice versa.
There are 2 such pairs that multiply together to form a third power of x here:
\(4x^2*3x +3x^2 *(-7x) = 12x^3 -21x^3 = -9x^3\)
Thus, we have our x3 coefficient for our first and second case, so all we have to do is now add them up.
\(-9x^3 +(-6x^3) = -15x^3\), giving us a coefficient of -15