Find f(g(x)) f(x)=x^2-5x , g(x)=x^2-7x+10

$f (x) = x^{2} - 5 x$ , $g (x) = x^{2} - 7 x + 10$

Set up the composite result function.

$f (g (x))$

Evaluate $f (x^{2} - 7 x + 10)$ by substituting in the value of $g$ into $f$ .

$f (x^{2} - 7 x + 10) = {(x^{2} - 7 x + 10)}^{2} - 5 (x^{2} - 7 x + 10)$

Simplify each term.

Rewrite ${(x^{2} - 7 x + 10)}^{2}$ as $(x^{2} - 7 x + 10) (x^{2} - 7 x + 10)$ .

$f (x^{2} - 7 x + 10) = (x^{2} - 7 x + 10) (x^{2} - 7 x + 10) - 5 (x^{2} - 7 x + 10)$

Expand $(x^{2} - 7 x + 10) (x^{2} - 7 x + 10)$ by multiplying each term in the first expression by each term in the second expression.

$f (x^{2} - 7 x + 10) = x^{2} x^{2} + x^{2} (- 7 x) + x^{2} \cdot 10 - 7 x \cdot x^{2} - 7 x (- 7 x) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

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.

$f (x^{2} - 7 x + 10) = x^{2 + 2} + x^{2} (- 7 x) + x^{2} \cdot 10 - 7 x \cdot x^{2} - 7 x (- 7 x) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

Add $2$ and $2$ .

$f (x^{2} - 7 x + 10) = x^{4} + x^{2} (- 7 x) + x^{2} \cdot 10 - 7 x \cdot x^{2} - 7 x (- 7 x) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

Rewrite using the commutative property of multiplication.

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{2} x + x^{2} \cdot 10 - 7 x \cdot x^{2} - 7 x (- 7 x) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

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

Move $x$ .

$f (x^{2} - 7 x + 10) = x^{4} - 7 (x \cdot x^{2}) + x^{2} \cdot 10 - 7 x \cdot x^{2} - 7 x (- 7 x) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

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

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

$f (x^{2} - 7 x + 10) = x^{4} - 7 (x \cdot x^{2}) + x^{2} \cdot 10 - 7 x \cdot x^{2} - 7 x (- 7 x) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

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

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{1 + 2} + x^{2} \cdot 10 - 7 x \cdot x^{2} - 7 x (- 7 x) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

Add $1$ and $2$ .

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{3} + x^{2} \cdot 10 - 7 x \cdot x^{2} - 7 x (- 7 x) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

Move $10$ to the left of $x^{2}$ .

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{3} + 10 \cdot x^{2} - 7 x \cdot x^{2} - 7 x (- 7 x) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

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

Move $x^{2}$ .

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{3} + 10 x^{2} - 7 (x^{2} x) - 7 x (- 7 x) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

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

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

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{3} + 10 x^{2} - 7 (x^{2} x) - 7 x (- 7 x) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

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

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{3} + 10 x^{2} - 7 x^{2 + 1} - 7 x (- 7 x) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

Add $2$ and $1$ .

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{3} + 10 x^{2} - 7 x^{3} - 7 x (- 7 x) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

Rewrite using the commutative property of multiplication.

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{3} + 10 x^{2} - 7 x^{3} - 7 \cdot (- 7 x \cdot x) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

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

Move $x$ .

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{3} + 10 x^{2} - 7 x^{3} - 7 \cdot (- 7 (x \cdot x)) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

Multiply $x$ by $x$ .

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{3} + 10 x^{2} - 7 x^{3} - 7 \cdot (- 7 x^{2}) - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

Multiply $- 7$ by $- 7$ .

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{3} + 10 x^{2} - 7 x^{3} + 49 x^{2} - 7 x \cdot 10 + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

Multiply $10$ by $- 7$ .

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{3} + 10 x^{2} - 7 x^{3} + 49 x^{2} - 70 x + 10 x^{2} + 10 (- 7 x) + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

Multiply $- 7$ by $10$ .

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{3} + 10 x^{2} - 7 x^{3} + 49 x^{2} - 70 x + 10 x^{2} - 70 x + 10 \cdot 10 - 5 (x^{2} - 7 x + 10)$

Multiply $10$ by $10$ .

$f (x^{2} - 7 x + 10) = x^{4} - 7 x^{3} + 10 x^{2} - 7 x^{3} + 49 x^{2} - 70 x + 10 x^{2} - 70 x + 100 - 5 (x^{2} - 7 x + 10)$

Subtract $7 x^{3}$ from $- 7 x^{3}$ .

$f (x^{2} - 7 x + 10) = x^{4} - 14 x^{3} + 10 x^{2} + 49 x^{2} - 70 x + 10 x^{2} - 70 x + 100 - 5 (x^{2} - 7 x + 10)$

Add $10 x^{2}$ and $49 x^{2}$ .

$f (x^{2} - 7 x + 10) = x^{4} - 14 x^{3} + 59 x^{2} - 70 x + 10 x^{2} - 70 x + 100 - 5 (x^{2} - 7 x + 10)$

Add $59 x^{2}$ and $10 x^{2}$ .

$f (x^{2} - 7 x + 10) = x^{4} - 14 x^{3} + 69 x^{2} - 70 x - 70 x + 100 - 5 (x^{2} - 7 x + 10)$

Subtract $70 x$ from $- 70 x$ .

$f (x^{2} - 7 x + 10) = x^{4} - 14 x^{3} + 69 x^{2} - 140 x + 100 - 5 (x^{2} - 7 x + 10)$

Apply the distributive property.

$f (x^{2} - 7 x + 10) = x^{4} - 14 x^{3} + 69 x^{2} - 140 x + 100 - 5 x^{2} - 5 (- 7 x) - 5 \cdot 10$

Simplify.

Multiply $- 7$ by $- 5$ .

$f (x^{2} - 7 x + 10) = x^{4} - 14 x^{3} + 69 x^{2} - 140 x + 100 - 5 x^{2} + 35 x - 5 \cdot 10$

Multiply $- 5$ by $10$ .

$f (x^{2} - 7 x + 10) = x^{4} - 14 x^{3} + 69 x^{2} - 140 x + 100 - 5 x^{2} + 35 x - 50$

Simplify by adding terms.

Subtract $5 x^{2}$ from $69 x^{2}$ .

$f (x^{2} - 7 x + 10) = x^{4} - 14 x^{3} + 64 x^{2} - 140 x + 100 + 35 x - 50$

Add $- 140 x$ and $35 x$ .

$f (x^{2} - 7 x + 10) = x^{4} - 14 x^{3} + 64 x^{2} - 105 x + 100 - 50$

Subtract $50$ from $100$ .

$f (x^{2} - 7 x + 10) = x^{4} - 14 x^{3} + 64 x^{2} - 105 x + 50$

Find f(g(x)) f(x)=x^2-5x , g(x)=x^2-7x+10

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