Find f(g(x)) f(x)=3x^3-2x^2+7x-5 g(x)=5x^2+x-1

$f (x) = 3 x^{3} - 2 x^{2} + 7 x - 5$ $g (x) = 5 x^{2} + x - 1$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (5 x^{2} + x - 1)$ by substituting in the value of $g$ into $f$ .

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

Step 3

Simplify each term.

Use the Multinomial Theorem.

$f (5 x^{2} + x - 1) = 3 ({(5 x^{2})}^{3} + 3 {(5 x^{2})}^{2} x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Simplify each term.

Apply the product rule to $5 x^{2}$ .

$f (5 x^{2} + x - 1) = 3 (5^{3} {(x^{2})}^{3} + 3 {(5 x^{2})}^{2} x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Raise $5$ to the power of $3$ .

$f (5 x^{2} + x - 1) = 3 (125 {(x^{2})}^{3} + 3 {(5 x^{2})}^{2} x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply the exponents in ${(x^{2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{2 \cdot 3} + 3 {(5 x^{2})}^{2} x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $2$ by $3$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 {(5 x^{2})}^{2} x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Apply the product rule to $5 x^{2}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 (5^{2} {(x^{2})}^{2}) x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Raise $5$ to the power of $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 (25 {(x^{2})}^{2}) x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 (25 x^{2 \cdot 2}) x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $2$ by $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 (25 x^{4}) x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $x^{4}$ by $x$ by adding the exponents.

Move $x$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 (25 (x \cdot x^{4})) + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

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

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

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

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 (25 x^{1 + 4}) + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Add $1$ and $4$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 (25 x^{5}) + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $25$ by $3$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

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

Move $x^{2}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 3 (5 (x^{2} x^{2})) + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

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

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 3 (5 x^{2 + 2}) + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Add $2$ and $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 3 (5 x^{4}) + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $5$ by $3$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Apply the product rule to $5 x^{2}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} + 3 (5^{2} {(x^{2})}^{2}) \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Raise $5$ to the power of $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} + 3 (25 {(x^{2})}^{2}) \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} + 3 (25 x^{2 \cdot 2}) \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $2$ by $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} + 3 (25 x^{4}) \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $25$ by $3$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} + 75 x^{4} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $- 1$ by $75$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

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

Move $x$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} + 6 (5 (x \cdot x^{2})) \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

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

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

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

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} + 6 (5 x^{1 + 2}) \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Add $1$ and $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} + 6 (5 x^{3}) \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $5$ by $6$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} + 30 x^{3} \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $- 1$ by $30$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $- 1$ by $3$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} - 3 x^{2} + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $5$ by $3$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} - 3 x^{2} + 15 x^{2} {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Raise $- 1$ to the power of $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} - 3 x^{2} + 15 x^{2} \cdot 1 + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $15$ by $1$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} - 3 x^{2} + 15 x^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Raise $- 1$ to the power of $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} - 3 x^{2} + 15 x^{2} + 3 x \cdot 1 + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $3$ by $1$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} - 3 x^{2} + 15 x^{2} + 3 x + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Raise $- 1$ to the power of $3$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} - 3 x^{2} + 15 x^{2} + 3 x - 1) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Subtract $75 x^{4}$ from $15 x^{4}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} - 60 x^{4} + x^{3} - 30 x^{3} - 3 x^{2} + 15 x^{2} + 3 x - 1) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Subtract $30 x^{3}$ from $x^{3}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} - 60 x^{4} - 29 x^{3} - 3 x^{2} + 15 x^{2} + 3 x - 1) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Add $- 3 x^{2}$ and $15 x^{2}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} - 60 x^{4} - 29 x^{3} + 12 x^{2} + 3 x - 1) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Apply the distributive property.

$f (5 x^{2} + x - 1) = 3 (125 x^{6}) + 3 (75 x^{5}) + 3 (- 60 x^{4}) + 3 (- 29 x^{3}) + 3 (12 x^{2}) + 3 (3 x) + 3 \cdot - 1 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Simplify.

Multiply $125$ by $3$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 3 (75 x^{5}) + 3 (- 60 x^{4}) + 3 (- 29 x^{3}) + 3 (12 x^{2}) + 3 (3 x) + 3 \cdot - 1 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $75$ by $3$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} + 3 (- 60 x^{4}) + 3 (- 29 x^{3}) + 3 (12 x^{2}) + 3 (3 x) + 3 \cdot - 1 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $- 60$ by $3$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} + 3 (- 29 x^{3}) + 3 (12 x^{2}) + 3 (3 x) + 3 \cdot - 1 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $- 29$ by $3$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 3 (12 x^{2}) + 3 (3 x) + 3 \cdot - 1 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $12$ by $3$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 3 (3 x) + 3 \cdot - 1 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $3$ by $3$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x + 3 \cdot - 1 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Multiply $3$ by $- 1$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Rewrite ${(5 x^{2} + x - 1)}^{2}$ as $(5 x^{2} + x - 1) (5 x^{2} + x - 1)$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 ((5 x^{2} + x - 1) (5 x^{2} + x - 1)) + 7 (5 x^{2} + x - 1) - 5$

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

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (5 x^{2} (5 x^{2}) + 5 x^{2} x + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Simplify each term.

Rewrite using the commutative property of multiplication.

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (5 \cdot (5 x^{2} x^{2}) + 5 x^{2} x + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

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

Move $x^{2}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (5 \cdot (5 (x^{2} x^{2})) + 5 x^{2} x + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

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

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (5 \cdot (5 x^{2 + 2}) + 5 x^{2} x + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Add $2$ and $2$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (5 \cdot (5 x^{4}) + 5 x^{2} x + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Multiply $5$ by $5$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{2} x + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

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

Move $x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 (x \cdot x^{2}) + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

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

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

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

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{1 + 2} + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Add $1$ and $2$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Multiply $- 1$ by $5$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Rewrite using the commutative property of multiplication.

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x \cdot x^{2} + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

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

Move $x^{2}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 (x^{2} x) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

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

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

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

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{2 + 1} + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Add $2$ and $1$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{3} + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Multiply $x$ by $x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{3} + x^{2} + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Move $- 1$ to the left of $x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{3} + x^{2} - 1 \cdot x - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Rewrite $- 1 x$ as $- x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{3} + x^{2} - x - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Multiply $5$ by $- 1$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{3} + x^{2} - x - 5 x^{2} - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Rewrite $- 1 x$ as $- x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{3} + x^{2} - x - 5 x^{2} - x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Multiply $- 1$ by $- 1$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{3} + x^{2} - x - 5 x^{2} - x + 1) + 7 (5 x^{2} + x - 1) - 5$

Add $5 x^{3}$ and $5 x^{3}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 10 x^{3} - 5 x^{2} + x^{2} - x - 5 x^{2} - x + 1) + 7 (5 x^{2} + x - 1) - 5$

Add $- 5 x^{2}$ and $x^{2}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 10 x^{3} - 4 x^{2} - x - 5 x^{2} - x + 1) + 7 (5 x^{2} + x - 1) - 5$

Subtract $5 x^{2}$ from $- 4 x^{2}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 10 x^{3} - 9 x^{2} - x - x + 1) + 7 (5 x^{2} + x - 1) - 5$

Subtract $x$ from $- x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 10 x^{3} - 9 x^{2} - 2 x + 1) + 7 (5 x^{2} + x - 1) - 5$

Apply the distributive property.

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4}) - 2 (10 x^{3}) - 2 (- 9 x^{2}) - 2 (- 2 x) - 2 \cdot 1 + 7 (5 x^{2} + x - 1) - 5$

Simplify.

Multiply $25$ by $- 2$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 2 (10 x^{3}) - 2 (- 9 x^{2}) - 2 (- 2 x) - 2 \cdot 1 + 7 (5 x^{2} + x - 1) - 5$

Multiply $10$ by $- 2$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 20 x^{3} - 2 (- 9 x^{2}) - 2 (- 2 x) - 2 \cdot 1 + 7 (5 x^{2} + x - 1) - 5$

Multiply $- 9$ by $- 2$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 20 x^{3} + 18 x^{2} - 2 (- 2 x) - 2 \cdot 1 + 7 (5 x^{2} + x - 1) - 5$

Multiply $- 2$ by $- 2$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 20 x^{3} + 18 x^{2} + 4 x - 2 \cdot 1 + 7 (5 x^{2} + x - 1) - 5$

Multiply $- 2$ by $1$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 20 x^{3} + 18 x^{2} + 4 x - 2 + 7 (5 x^{2} + x - 1) - 5$

Apply the distributive property.

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 20 x^{3} + 18 x^{2} + 4 x - 2 + 7 (5 x^{2}) + 7 x + 7 \cdot - 1 - 5$

Simplify.

Multiply $5$ by $7$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 20 x^{3} + 18 x^{2} + 4 x - 2 + 35 x^{2} + 7 x + 7 \cdot - 1 - 5$

Multiply $7$ by $- 1$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 20 x^{3} + 18 x^{2} + 4 x - 2 + 35 x^{2} + 7 x - 7 - 5$

Step 4

Simplify by adding terms.

Subtract $50 x^{4}$ from $- 180 x^{4}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 20 x^{3} + 18 x^{2} + 4 x - 2 + 35 x^{2} + 7 x - 7 - 5$

Subtract $20 x^{3}$ from $- 87 x^{3}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 36 x^{2} + 9 x - 3 + 18 x^{2} + 4 x - 2 + 35 x^{2} + 7 x - 7 - 5$

Add $36 x^{2}$ and $18 x^{2}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 54 x^{2} + 9 x - 3 + 4 x - 2 + 35 x^{2} + 7 x - 7 - 5$

Add $54 x^{2}$ and $35 x^{2}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 89 x^{2} + 9 x - 3 + 4 x - 2 + 7 x - 7 - 5$

Add $9 x$ and $4 x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 89 x^{2} + 13 x - 3 - 2 + 7 x - 7 - 5$

Add $13 x$ and $7 x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 89 x^{2} + 20 x - 3 - 2 - 7 - 5$

Simplify by subtracting numbers.

Subtract $2$ from $- 3$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 89 x^{2} + 20 x - 5 - 7 - 5$

Subtract $7$ from $- 5$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 89 x^{2} + 20 x - 12 - 5$

Subtract $5$ from $- 12$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 89 x^{2} + 20 x - 17$

Find f(g(x)) f(x)=3x^3-2x^2+7x-5 g(x)=5x^2+x-1

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