Find the Product (3f-2g)(f^2+2(fg)-g^2)

$(3 f - 2 g) (f^{2} + 2 (f g) - g^{2})$

Step 1

Expand $(3 f - 2 g) (f^{2} + 2 f g - g^{2})$ by multiplying each term in the first expression by each term in the second expression.

$3 f \cdot f^{2} + 3 f (2 f g) + 3 f (- g^{2}) - 2 g f^{2} - 2 g (2 f g) - 2 g (- g^{2})$

Step 2

Simplify each term.

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

Move $f^{2}$ .

$3 (f^{2} f) + 3 f (2 f g) + 3 f (- g^{2}) - 2 g f^{2} - 2 g (2 f g) - 2 g (- g^{2})$

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

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

$3 (f^{2} f^{1}) + 3 f (2 f g) + 3 f (- g^{2}) - 2 g f^{2} - 2 g (2 f g) - 2 g (- g^{2})$

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

$3 f^{2 + 1} + 3 f (2 f g) + 3 f (- g^{2}) - 2 g f^{2} - 2 g (2 f g) - 2 g (- g^{2})$

Add $2$ and $1$ .

$3 f^{3} + 3 f (2 f g) + 3 f (- g^{2}) - 2 g f^{2} - 2 g (2 f g) - 2 g (- g^{2})$

Multiply $f$ by $f$ by adding the exponents.

Move $f$ .

$3 f^{3} + 3 (f \cdot f) (2 g) + 3 f (- g^{2}) - 2 g f^{2} - 2 g (2 f g) - 2 g (- g^{2})$

Multiply $f$ by $f$ .

$3 f^{3} + 3 f^{2} (2 g) + 3 f (- g^{2}) - 2 g f^{2} - 2 g (2 f g) - 2 g (- g^{2})$

Rewrite using the commutative property of multiplication.

$3 f^{3} + 3 \cdot 2 f^{2} g + 3 f (- g^{2}) - 2 g f^{2} - 2 g (2 f g) - 2 g (- g^{2})$

Multiply $3$ by $2$ .

$3 f^{3} + 6 f^{2} g + 3 f (- g^{2}) - 2 g f^{2} - 2 g (2 f g) - 2 g (- g^{2})$

Rewrite using the commutative property of multiplication.

$3 f^{3} + 6 f^{2} g + 3 \cdot - 1 f g^{2} - 2 g f^{2} - 2 g (2 f g) - 2 g (- g^{2})$

Multiply $3$ by $- 1$ .

$3 f^{3} + 6 f^{2} g - 3 f g^{2} - 2 g f^{2} - 2 g (2 f g) - 2 g (- g^{2})$

Multiply $g$ by $g$ by adding the exponents.

Move $g$ .

$3 f^{3} + 6 f^{2} g - 3 f g^{2} - 2 g f^{2} - 2 (g \cdot g) (2 f) - 2 g (- g^{2})$

Multiply $g$ by $g$ .

$3 f^{3} + 6 f^{2} g - 3 f g^{2} - 2 g f^{2} - 2 g^{2} (2 f) - 2 g (- g^{2})$

Rewrite using the commutative property of multiplication.

$3 f^{3} + 6 f^{2} g - 3 f g^{2} - 2 g f^{2} - 2 \cdot 2 g^{2} f - 2 g (- g^{2})$

Multiply $- 2$ by $2$ .

$3 f^{3} + 6 f^{2} g - 3 f g^{2} - 2 g f^{2} - 4 g^{2} f - 2 g (- g^{2})$

Rewrite using the commutative property of multiplication.

$3 f^{3} + 6 f^{2} g - 3 f g^{2} - 2 g f^{2} - 4 g^{2} f - 2 \cdot - 1 g \cdot g^{2}$

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

Move $g^{2}$ .

$3 f^{3} + 6 f^{2} g - 3 f g^{2} - 2 g f^{2} - 4 g^{2} f - 2 \cdot - 1 (g^{2} g)$

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

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

$3 f^{3} + 6 f^{2} g - 3 f g^{2} - 2 g f^{2} - 4 g^{2} f - 2 \cdot - 1 (g^{2} g^{1})$

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

$3 f^{3} + 6 f^{2} g - 3 f g^{2} - 2 g f^{2} - 4 g^{2} f - 2 \cdot - 1 g^{2 + 1}$

Add $2$ and $1$ .

$3 f^{3} + 6 f^{2} g - 3 f g^{2} - 2 g f^{2} - 4 g^{2} f - 2 \cdot - 1 g^{3}$

Multiply $- 2$ by $- 1$ .

$3 f^{3} + 6 f^{2} g - 3 f g^{2} - 2 g f^{2} - 4 g^{2} f + 2 g^{3}$

Step 3

Subtract $2 g f^{2}$ from $6 f^{2} g$ .

Move $g$ .

$3 f^{3} + 6 f^{2} g - 2 f^{2} g - 3 f g^{2} - 4 g^{2} f + 2 g^{3}$

Subtract $2 f^{2} g$ from $6 f^{2} g$ .

$3 f^{3} + 4 f^{2} g - 3 f g^{2} - 4 g^{2} f + 2 g^{3}$

Step 4

Subtract $4 g^{2} f$ from $- 3 f g^{2}$ .

Move $g^{2}$ .

$3 f^{3} + 4 f^{2} g - 3 f g^{2} - 4 f g^{2} + 2 g^{3}$

Subtract $4 f g^{2}$ from $- 3 f g^{2}$ .

$3 f^{3} + 4 f^{2} g - 7 f g^{2} + 2 g^{3}$

Find the Product (3f-2g)(f^2+2(fg)-g^2)

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