Question

f(x)·g(x)
x² -4x -5
x² x⁴ -4x³
-2x
-15

Explanation:

Step1: Multiply \(x^2\) with each term

We have the first row (from \(x^2\)) already filled for \(x^2\times x^2 = x^4\), \(x^2\times(-4x)=-4x^3\), and \(x^2\times(-5)=-5x^2\).

Step2: Multiply \(-2x\) with each term

\(-2x\times x^2=-2x^3\), \(-2x\times(-4x) = 8x^2\), \(-2x\times(-5)=10x\).

Step3: Multiply \(-15\) with each term

\(-15\times x^2=-15x^2\), \(-15\times(-4x)=60x\), \(-15\times(-5) = 75\).

Now let's fill the table:

	\(x^2\)	\(-4x\)	\(-5\)
\(-2x\)	\(-2x^3\)	\(8x^2\)	\(10x\)
\(-15\)	\(-15x^2\)	\(60x\)	\(75\)

After filling, we can also expand \((x^2 - 2x - 15)(x^2 - 4x - 5)\) by adding all the terms:

\(x^4-4x^3 - 5x^2-2x^3 + 8x^2+10x-15x^2+60x + 75\)

Combine like terms:

\(x^4+(-4x^3-2x^3)+(-5x^2 + 8x^2-15x^2)+(10x + 60x)+75\)

\(x^4-6x^3-12x^2 + 70x+75\)

But if the task is just to fill the table, the filled cells are as above.

Answer:

The filled table (for the missing cells) is:

In the row of \(-2x\): \(-2x^3\), \(8x^2\), \(10x\)

In the row of \(-15\): \(-15x^2\), \(60x\), \(75\)

And in the column of \(-5\) (first row): \(-5x^2\)