Find the Product f(x)=-x^2 , g(x)=4x-x^2

$f (x) = - x^{2}$ , $g (x) = 4 x - x^{2}$

Step 1

Replace the function designators with the actual functions in $f (x) \cdot (g (x))$ .

$(- x^{2}) \cdot (4 x - x^{2})$

Step 2

Simplify.

Simplify by multiplying through.

Apply the distributive property.

$- x^{2} (4 x) - x^{2} (- x^{2})$

Reorder.

Rewrite using the commutative property of multiplication.

$- 1 \cdot 4 x^{2} x - x^{2} (- x^{2})$

Rewrite using the commutative property of multiplication.

$- 1 \cdot 4 x^{2} x - 1 \cdot - 1 x^{2} x^{2}$

Simplify each term.

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$- 1 \cdot 4 (x \cdot x^{2}) - 1 \cdot - 1 x^{2} x^{2}$

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$- 1 \cdot 4 (x^{1} x^{2}) - 1 \cdot - 1 x^{2} x^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 1 \cdot 4 x^{1 + 2} - 1 \cdot - 1 x^{2} x^{2}$

Add $1$ and $2$ .

$- 1 \cdot 4 x^{3} - 1 \cdot - 1 x^{2} x^{2}$

Multiply $- 1$ by $4$ .

$- 4 x^{3} - 1 \cdot - 1 x^{2} x^{2}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$- 4 x^{3} - 1 \cdot - 1 (x^{2} x^{2})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 4 x^{3} - 1 \cdot - 1 x^{2 + 2}$

Add $2$ and $2$ .

$- 4 x^{3} - 1 \cdot - 1 x^{4}$

Multiply $- 1$ by $- 1$ .

$- 4 x^{3} + 1 x^{4}$

Multiply $x^{4}$ by $1$ .

$- 4 x^{3} + x^{4}$