Find the Product f(x)=x^2-2x-5 , g(x)=x^2-2x+1

$f (x) = x^{2} - 2 x - 5$ , $g (x) = x^{2} - 2 x + 1$

Step 1

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

$(x^{2} - 2 x - 5) \cdot (x^{2} - 2 x + 1)$

Step 2

Simplify.

Expand $(x^{2} - 2 x - 5) (x^{2} - 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} x^{2} + x^{2} (- 2 x) + x^{2} \cdot 1 - 2 x \cdot x^{2} - 2 x (- 2 x) - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

Simplify terms.

Simplify each term.

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

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

$x^{2 + 2} + x^{2} (- 2 x) + x^{2} \cdot 1 - 2 x \cdot x^{2} - 2 x (- 2 x) - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

Add $2$ and $2$ .

$x^{4} + x^{2} (- 2 x) + x^{2} \cdot 1 - 2 x \cdot x^{2} - 2 x (- 2 x) - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{4} - 2 x^{2} x + x^{2} \cdot 1 - 2 x \cdot x^{2} - 2 x (- 2 x) - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

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

Move $x$ .

$x^{4} - 2 (x \cdot x^{2}) + x^{2} \cdot 1 - 2 x \cdot x^{2} - 2 x (- 2 x) - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

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

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

$x^{4} - 2 (x^{1} x^{2}) + x^{2} \cdot 1 - 2 x \cdot x^{2} - 2 x (- 2 x) - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

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

$x^{4} - 2 x^{1 + 2} + x^{2} \cdot 1 - 2 x \cdot x^{2} - 2 x (- 2 x) - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

Add $1$ and $2$ .

$x^{4} - 2 x^{3} + x^{2} \cdot 1 - 2 x \cdot x^{2} - 2 x (- 2 x) - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

Multiply $x^{2}$ by $1$ .

$x^{4} - 2 x^{3} + x^{2} - 2 x \cdot x^{2} - 2 x (- 2 x) - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

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

Move $x^{2}$ .

$x^{4} - 2 x^{3} + x^{2} - 2 (x^{2} x) - 2 x (- 2 x) - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

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

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

$x^{4} - 2 x^{3} + x^{2} - 2 (x^{2} x^{1}) - 2 x (- 2 x) - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

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

$x^{4} - 2 x^{3} + x^{2} - 2 x^{2 + 1} - 2 x (- 2 x) - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

Add $2$ and $1$ .

$x^{4} - 2 x^{3} + x^{2} - 2 x^{3} - 2 x (- 2 x) - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{4} - 2 x^{3} + x^{2} - 2 x^{3} - 2 \cdot - 2 x \cdot x - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x^{4} - 2 x^{3} + x^{2} - 2 x^{3} - 2 \cdot - 2 (x \cdot x) - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

Multiply $x$ by $x$ .

$x^{4} - 2 x^{3} + x^{2} - 2 x^{3} - 2 \cdot - 2 x^{2} - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

Multiply $- 2$ by $- 2$ .

$x^{4} - 2 x^{3} + x^{2} - 2 x^{3} + 4 x^{2} - 2 x \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

Multiply $- 2$ by $1$ .

$x^{4} - 2 x^{3} + x^{2} - 2 x^{3} + 4 x^{2} - 2 x - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1$

Multiply $- 2$ by $- 5$ .

$x^{4} - 2 x^{3} + x^{2} - 2 x^{3} + 4 x^{2} - 2 x - 5 x^{2} + 10 x - 5 \cdot 1$

Multiply $- 5$ by $1$ .

$x^{4} - 2 x^{3} + x^{2} - 2 x^{3} + 4 x^{2} - 2 x - 5 x^{2} + 10 x - 5$

Simplify by adding terms.

Subtract $2 x^{3}$ from $- 2 x^{3}$ .

$x^{4} - 4 x^{3} + x^{2} + 4 x^{2} - 2 x - 5 x^{2} + 10 x - 5$

Add $x^{2}$ and $4 x^{2}$ .

$x^{4} - 4 x^{3} + 5 x^{2} - 2 x - 5 x^{2} + 10 x - 5$

Combine the opposite terms in $x^{4} - 4 x^{3} + 5 x^{2} - 2 x - 5 x^{2} + 10 x - 5$ .

Subtract $5 x^{2}$ from $5 x^{2}$ .

$x^{4} - 4 x^{3} - 2 x + 0 + 10 x - 5$

Add $x^{4} - 4 x^{3} - 2 x$ and $0$ .

$x^{4} - 4 x^{3} - 2 x + 10 x - 5$

Add $- 2 x$ and $10 x$ .

$x^{4} - 4 x^{3} + 8 x - 5$

Find the Product f(x)=x^2-2x-5 , g(x)=x^2-2x+1

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