Find g(h(x)) g(x)=x^2+6x h(x)=x^2-1

$g (x) = x^{2} + 6 x$ $h (x) = x^{2} - 1$

Step 1

Set up the composite result function.

$g (h (x))$

Step 2

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

$g (x^{2} - 1) = {(x^{2} - 1)}^{2} + 6 (x^{2} - 1)$

Step 3

Simplify each term.

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

$g (x^{2} - 1) = (x^{2} - 1) (x^{2} - 1) + 6 (x^{2} - 1)$

Expand $(x^{2} - 1) (x^{2} - 1)$ using the FOIL Method.

Apply the distributive property.

$g (x^{2} - 1) = x^{2} (x^{2} - 1) - 1 (x^{2} - 1) + 6 (x^{2} - 1)$

Apply the distributive property.

$g (x^{2} - 1) = x^{2} x^{2} + x^{2} \cdot - 1 - 1 (x^{2} - 1) + 6 (x^{2} - 1)$

Apply the distributive property.

$g (x^{2} - 1) = x^{2} x^{2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1 + 6 (x^{2} - 1)$

Simplify and combine like 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.

$g (x^{2} - 1) = x^{2 + 2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1 + 6 (x^{2} - 1)$

Add $2$ and $2$ .

$g (x^{2} - 1) = x^{4} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1 + 6 (x^{2} - 1)$

Move $- 1$ to the left of $x^{2}$ .

$g (x^{2} - 1) = x^{4} - 1 \cdot x^{2} - 1 x^{2} - 1 \cdot - 1 + 6 (x^{2} - 1)$

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$g (x^{2} - 1) = x^{4} - x^{2} - 1 x^{2} - 1 \cdot - 1 + 6 (x^{2} - 1)$

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$g (x^{2} - 1) = x^{4} - x^{2} - x^{2} - 1 \cdot - 1 + 6 (x^{2} - 1)$

Multiply $- 1$ by $- 1$ .

$g (x^{2} - 1) = x^{4} - x^{2} - x^{2} + 1 + 6 (x^{2} - 1)$

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

$g (x^{2} - 1) = x^{4} - 2 x^{2} + 1 + 6 (x^{2} - 1)$

Apply the distributive property.

$g (x^{2} - 1) = x^{4} - 2 x^{2} + 1 + 6 x^{2} + 6 \cdot - 1$

Multiply $6$ by $- 1$ .

$g (x^{2} - 1) = x^{4} - 2 x^{2} + 1 + 6 x^{2} - 6$

Step 4

Simplify by adding terms.

Add $- 2 x^{2}$ and $6 x^{2}$ .

$g (x^{2} - 1) = x^{4} + 4 x^{2} + 1 - 6$

Subtract $6$ from $1$ .

$g (x^{2} - 1) = x^{4} + 4 x^{2} - 5$