Calculus Examples

$y = {(3 x + 4)}^{2} {(x - 5)}^{3}$

Step 1

Rewrite ${(3 x + 4)}^{2}$ as $(3 x + 4) (3 x + 4)$ .

$\frac{d}{d x} [(3 x + 4) (3 x + 4) {(x - 5)}^{3}]$

Step 2

Expand $(3 x + 4) (3 x + 4)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [(3 x (3 x + 4) + 4 (3 x + 4)) {(x - 5)}^{3}]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [(3 x (3 x) + 3 x \cdot 4 + 4 (3 x + 4)) {(x - 5)}^{3}]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [(3 x (3 x) + 3 x \cdot 4 + 4 (3 x) + 4 \cdot 4) {(x - 5)}^{3}]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [(3 \cdot 3 x \cdot x + 3 x \cdot 4 + 4 (3 x) + 4 \cdot 4) {(x - 5)}^{3}]$

Step 3.1.2

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

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Step 3.1.2.1

Move $x$ .

$\frac{d}{d x} [(3 \cdot 3 (x \cdot x) + 3 x \cdot 4 + 4 (3 x) + 4 \cdot 4) {(x - 5)}^{3}]$

Step 3.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [(3 \cdot 3 x^{2} + 3 x \cdot 4 + 4 (3 x) + 4 \cdot 4) {(x - 5)}^{3}]$

Step 3.1.3

Multiply $3$ by $3$ .

$\frac{d}{d x} [(9 x^{2} + 3 x \cdot 4 + 4 (3 x) + 4 \cdot 4) {(x - 5)}^{3}]$

Step 3.1.4

Multiply $4$ by $3$ .

$\frac{d}{d x} [(9 x^{2} + 12 x + 4 (3 x) + 4 \cdot 4) {(x - 5)}^{3}]$

Step 3.1.5

Multiply $3$ by $4$ .

$\frac{d}{d x} [(9 x^{2} + 12 x + 12 x + 4 \cdot 4) {(x - 5)}^{3}]$

Step 3.1.6

Multiply $4$ by $4$ .

$\frac{d}{d x} [(9 x^{2} + 12 x + 12 x + 16) {(x - 5)}^{3}]$

Step 3.2

Add $12 x$ and $12 x$ .

$\frac{d}{d x} [(9 x^{2} + 24 x + 16) {(x - 5)}^{3}]$

Step 4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 9 x^{2} + 24 x + 16$ and $g (x) = {(x - 5)}^{3}$ .

$(9 x^{2} + 24 x + 16) \frac{d}{d x} [{(x - 5)}^{3}] + {(x - 5)}^{3} \frac{d}{d x} [9 x^{2} + 24 x + 16]$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x - 5$ .

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Step 5.1

To apply the Chain Rule, set $u$ as $x - 5$ .

$(9 x^{2} + 24 x + 16) (\frac{d}{d u} [u^{3}] \frac{d}{d x} [x - 5]) + {(x - 5)}^{3} \frac{d}{d x} [9 x^{2} + 24 x + 16]$

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$(9 x^{2} + 24 x + 16) (3 u^{2} \frac{d}{d x} [x - 5]) + {(x - 5)}^{3} \frac{d}{d x} [9 x^{2} + 24 x + 16]$

Step 5.3

Replace all occurrences of $u$ with $x - 5$ .

$(9 x^{2} + 24 x + 16) (3 {(x - 5)}^{2} \frac{d}{d x} [x - 5]) + {(x - 5)}^{3} \frac{d}{d x} [9 x^{2} + 24 x + 16]$

Step 6

Differentiate.

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Step 6.1

Move $3$ to the left of $9 x^{2} + 24 x + 16$ .

$3 \cdot (9 x^{2} + 24 x + 16) {(x - 5)}^{2} \frac{d}{d x} [x - 5] + {(x - 5)}^{3} \frac{d}{d x} [9 x^{2} + 24 x + 16]$

Step 6.2

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$3 (9 x^{2} + 24 x + 16) {(x - 5)}^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 5]) + {(x - 5)}^{3} \frac{d}{d x} [9 x^{2} + 24 x + 16]$

Step 6.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 (9 x^{2} + 24 x + 16) {(x - 5)}^{2} (1 + \frac{d}{d x} [- 5]) + {(x - 5)}^{3} \frac{d}{d x} [9 x^{2} + 24 x + 16]$

Step 6.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$3 (9 x^{2} + 24 x + 16) {(x - 5)}^{2} (1 + 0) + {(x - 5)}^{3} \frac{d}{d x} [9 x^{2} + 24 x + 16]$

Step 6.5

Simplify the expression.

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Step 6.5.1

Add $1$ and $0$ .

$3 (9 x^{2} + 24 x + 16) {(x - 5)}^{2} \cdot 1 + {(x - 5)}^{3} \frac{d}{d x} [9 x^{2} + 24 x + 16]$

Step 6.5.2

Multiply $3$ by $1$ .

$3 (9 x^{2} + 24 x + 16) {(x - 5)}^{2} + {(x - 5)}^{3} \frac{d}{d x} [9 x^{2} + 24 x + 16]$

Step 6.6

By the Sum Rule, the derivative of $9 x^{2} + 24 x + 16$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [24 x] + \frac{d}{d x} [16]$ .

$3 (9 x^{2} + 24 x + 16) {(x - 5)}^{2} + {(x - 5)}^{3} (\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [24 x] + \frac{d}{d x} [16])$

Step 6.7

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$3 (9 x^{2} + 24 x + 16) {(x - 5)}^{2} + {(x - 5)}^{3} (9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [24 x] + \frac{d}{d x} [16])$

Step 6.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$3 (9 x^{2} + 24 x + 16) {(x - 5)}^{2} + {(x - 5)}^{3} (9 (2 x) + \frac{d}{d x} [24 x] + \frac{d}{d x} [16])$

Step 6.9

Multiply $2$ by $9$ .

$3 (9 x^{2} + 24 x + 16) {(x - 5)}^{2} + {(x - 5)}^{3} (18 x + \frac{d}{d x} [24 x] + \frac{d}{d x} [16])$

Step 6.10

Since $24$ is constant with respect to $x$ , the derivative of $24 x$ with respect to $x$ is $24 \frac{d}{d x} [x]$ .

$3 (9 x^{2} + 24 x + 16) {(x - 5)}^{2} + {(x - 5)}^{3} (18 x + 24 \frac{d}{d x} [x] + \frac{d}{d x} [16])$

Step 6.11

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 (9 x^{2} + 24 x + 16) {(x - 5)}^{2} + {(x - 5)}^{3} (18 x + 24 \cdot 1 + \frac{d}{d x} [16])$

Step 6.12

Multiply $24$ by $1$ .

$3 (9 x^{2} + 24 x + 16) {(x - 5)}^{2} + {(x - 5)}^{3} (18 x + 24 + \frac{d}{d x} [16])$

Step 6.13

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$3 (9 x^{2} + 24 x + 16) {(x - 5)}^{2} + {(x - 5)}^{3} (18 x + 24 + 0)$

Step 6.14

Add $18 x + 24$ and $0$ .

$3 (9 x^{2} + 24 x + 16) {(x - 5)}^{2} + {(x - 5)}^{3} (18 x + 24)$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$(3 (9 x^{2}) + 3 (24 x) + 3 \cdot 16) {(x - 5)}^{2} + {(x - 5)}^{3} (18 x + 24)$

Step 7.2

Multiply $9$ by $3$ .

$(27 x^{2} + 3 (24 x) + 3 \cdot 16) {(x - 5)}^{2} + {(x - 5)}^{3} (18 x + 24)$

Step 7.3

Multiply $24$ by $3$ .

$(27 x^{2} + 72 x + 3 \cdot 16) {(x - 5)}^{2} + {(x - 5)}^{3} (18 x + 24)$

Step 7.4

Multiply $3$ by $16$ .

$(27 x^{2} + 72 x + 48) {(x - 5)}^{2} + {(x - 5)}^{3} (18 x + 24)$

Step 7.5

Factor ${(x - 5)}^{2}$ out of $(27 x^{2} + 72 x + 48) {(x - 5)}^{2} + {(x - 5)}^{3} (18 x + 24)$ .

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Step 7.5.1

Factor ${(x - 5)}^{2}$ out of $(27 x^{2} + 72 x + 48) {(x - 5)}^{2}$ .

${(x - 5)}^{2} (27 x^{2} + 72 x + 48) + {(x - 5)}^{3} (18 x + 24)$

Step 7.5.2

Factor ${(x - 5)}^{2}$ out of ${(x - 5)}^{3} (18 x + 24)$ .

${(x - 5)}^{2} (27 x^{2} + 72 x + 48) + {(x - 5)}^{2} ((x - 5) (18 x + 24))$

Step 7.5.3

Factor ${(x - 5)}^{2}$ out of ${(x - 5)}^{2} (27 x^{2} + 72 x + 48) + {(x - 5)}^{2} ((x - 5) (18 x + 24))$ .

${(x - 5)}^{2} (27 x^{2} + 72 x + 48 + (x - 5) (18 x + 24))$

Step 7.6

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

$(x - 5) (x - 5) (27 x^{2} + 72 x + 48 + (x - 5) (18 x + 24))$

Step 7.7

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 7.7.1

Apply the distributive property.

$(x (x - 5) - 5 (x - 5)) (27 x^{2} + 72 x + 48 + (x - 5) (18 x + 24))$

Step 7.7.2

Apply the distributive property.

$(x \cdot x + x \cdot - 5 - 5 (x - 5)) (27 x^{2} + 72 x + 48 + (x - 5) (18 x + 24))$

Step 7.7.3

Apply the distributive property.

$(x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) (27 x^{2} + 72 x + 48 + (x - 5) (18 x + 24))$

Step 7.8

Simplify and combine like terms.

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Step 7.8.1

Simplify each term.

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Step 7.8.1.1

Multiply $x$ by $x$ .

$(x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5) (27 x^{2} + 72 x + 48 + (x - 5) (18 x + 24))$

Step 7.8.1.2

Move $- 5$ to the left of $x$ .

$(x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5) (27 x^{2} + 72 x + 48 + (x - 5) (18 x + 24))$

Step 7.8.1.3

Multiply $- 5$ by $- 5$ .

$(x^{2} - 5 x - 5 x + 25) (27 x^{2} + 72 x + 48 + (x - 5) (18 x + 24))$

Step 7.8.2

Subtract $5 x$ from $- 5 x$ .

$(x^{2} - 10 x + 25) (27 x^{2} + 72 x + 48 + (x - 5) (18 x + 24))$

Step 7.9

Simplify each term.

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Step 7.9.1

Expand $(x - 5) (18 x + 24)$ using the FOIL Method.

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Step 7.9.1.1

Apply the distributive property.

$(x^{2} - 10 x + 25) (27 x^{2} + 72 x + 48 + x (18 x + 24) - 5 (18 x + 24))$

Step 7.9.1.2

Apply the distributive property.

$(x^{2} - 10 x + 25) (27 x^{2} + 72 x + 48 + x (18 x) + x \cdot 24 - 5 (18 x + 24))$

Step 7.9.1.3

Apply the distributive property.

$(x^{2} - 10 x + 25) (27 x^{2} + 72 x + 48 + x (18 x) + x \cdot 24 - 5 (18 x) - 5 \cdot 24)$

Step 7.9.2

Simplify and combine like terms.

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Step 7.9.2.1

Simplify each term.

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Step 7.9.2.1.1

Rewrite using the commutative property of multiplication.

$(x^{2} - 10 x + 25) (27 x^{2} + 72 x + 48 + 18 x \cdot x + x \cdot 24 - 5 (18 x) - 5 \cdot 24)$

Step 7.9.2.1.2

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

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Step 7.9.2.1.2.1

Move $x$ .

$(x^{2} - 10 x + 25) (27 x^{2} + 72 x + 48 + 18 (x \cdot x) + x \cdot 24 - 5 (18 x) - 5 \cdot 24)$

Step 7.9.2.1.2.2

Multiply $x$ by $x$ .

$(x^{2} - 10 x + 25) (27 x^{2} + 72 x + 48 + 18 x^{2} + x \cdot 24 - 5 (18 x) - 5 \cdot 24)$

Step 7.9.2.1.3

Move $24$ to the left of $x$ .

$(x^{2} - 10 x + 25) (27 x^{2} + 72 x + 48 + 18 x^{2} + 24 \cdot x - 5 (18 x) - 5 \cdot 24)$

Step 7.9.2.1.4

Multiply $18$ by $- 5$ .

$(x^{2} - 10 x + 25) (27 x^{2} + 72 x + 48 + 18 x^{2} + 24 x - 90 x - 5 \cdot 24)$

Step 7.9.2.1.5

Multiply $- 5$ by $24$ .

$(x^{2} - 10 x + 25) (27 x^{2} + 72 x + 48 + 18 x^{2} + 24 x - 90 x - 120)$

Step 7.9.2.2

Subtract $90 x$ from $24 x$ .

$(x^{2} - 10 x + 25) (27 x^{2} + 72 x + 48 + 18 x^{2} - 66 x - 120)$

Step 7.10

Add $27 x^{2}$ and $18 x^{2}$ .

$(x^{2} - 10 x + 25) (45 x^{2} + 72 x + 48 - 66 x - 120)$

Step 7.11

Subtract $66 x$ from $72 x$ .

$(x^{2} - 10 x + 25) (45 x^{2} + 6 x + 48 - 120)$

Step 7.12

Subtract $120$ from $48$ .

$(x^{2} - 10 x + 25) (45 x^{2} + 6 x - 72)$

Step 7.13

Expand $(x^{2} - 10 x + 25) (45 x^{2} + 6 x - 72)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} (45 x^{2}) + x^{2} (6 x) + x^{2} \cdot - 72 - 10 x (45 x^{2}) - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14

Simplify each term.

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Step 7.14.1

Rewrite using the commutative property of multiplication.

$45 x^{2} x^{2} + x^{2} (6 x) + x^{2} \cdot - 72 - 10 x (45 x^{2}) - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.2

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

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Step 7.14.2.1

Move $x^{2}$ .

$45 (x^{2} x^{2}) + x^{2} (6 x) + x^{2} \cdot - 72 - 10 x (45 x^{2}) - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.2.2

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

$45 x^{2 + 2} + x^{2} (6 x) + x^{2} \cdot - 72 - 10 x (45 x^{2}) - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.2.3

Add $2$ and $2$ .

$45 x^{4} + x^{2} (6 x) + x^{2} \cdot - 72 - 10 x (45 x^{2}) - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.3

Rewrite using the commutative property of multiplication.

$45 x^{4} + 6 x^{2} x + x^{2} \cdot - 72 - 10 x (45 x^{2}) - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.4

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

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Step 7.14.4.1

Move $x$ .

$45 x^{4} + 6 (x \cdot x^{2}) + x^{2} \cdot - 72 - 10 x (45 x^{2}) - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.4.2

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

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Step 7.14.4.2.1

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

$45 x^{4} + 6 (x^{1} x^{2}) + x^{2} \cdot - 72 - 10 x (45 x^{2}) - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.4.2.2

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

$45 x^{4} + 6 x^{1 + 2} + x^{2} \cdot - 72 - 10 x (45 x^{2}) - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.4.3

Add $1$ and $2$ .

$45 x^{4} + 6 x^{3} + x^{2} \cdot - 72 - 10 x (45 x^{2}) - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.5

Move $- 72$ to the left of $x^{2}$ .

$45 x^{4} + 6 x^{3} - 72 \cdot x^{2} - 10 x (45 x^{2}) - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.6

Rewrite using the commutative property of multiplication.

$45 x^{4} + 6 x^{3} - 72 x^{2} - 10 \cdot 45 x \cdot x^{2} - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.7

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

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Step 7.14.7.1

Move $x^{2}$ .

$45 x^{4} + 6 x^{3} - 72 x^{2} - 10 \cdot 45 (x^{2} x) - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.7.2

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

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Step 7.14.7.2.1

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

$45 x^{4} + 6 x^{3} - 72 x^{2} - 10 \cdot 45 (x^{2} x^{1}) - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.7.2.2

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

$45 x^{4} + 6 x^{3} - 72 x^{2} - 10 \cdot 45 x^{2 + 1} - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.7.3

Add $2$ and $1$ .

$45 x^{4} + 6 x^{3} - 72 x^{2} - 10 \cdot 45 x^{3} - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.8

Multiply $- 10$ by $45$ .

$45 x^{4} + 6 x^{3} - 72 x^{2} - 450 x^{3} - 10 x (6 x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.9

Rewrite using the commutative property of multiplication.

$45 x^{4} + 6 x^{3} - 72 x^{2} - 450 x^{3} - 10 \cdot 6 x \cdot x - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.10

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

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Step 7.14.10.1

Move $x$ .

$45 x^{4} + 6 x^{3} - 72 x^{2} - 450 x^{3} - 10 \cdot 6 (x \cdot x) - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.10.2

Multiply $x$ by $x$ .

$45 x^{4} + 6 x^{3} - 72 x^{2} - 450 x^{3} - 10 \cdot 6 x^{2} - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.11

Multiply $- 10$ by $6$ .

$45 x^{4} + 6 x^{3} - 72 x^{2} - 450 x^{3} - 60 x^{2} - 10 x \cdot - 72 + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.12

Multiply $- 72$ by $- 10$ .

$45 x^{4} + 6 x^{3} - 72 x^{2} - 450 x^{3} - 60 x^{2} + 720 x + 25 (45 x^{2}) + 25 (6 x) + 25 \cdot - 72$

Step 7.14.13

Multiply $45$ by $25$ .

$45 x^{4} + 6 x^{3} - 72 x^{2} - 450 x^{3} - 60 x^{2} + 720 x + 1125 x^{2} + 25 (6 x) + 25 \cdot - 72$

Step 7.14.14

Multiply $6$ by $25$ .

$45 x^{4} + 6 x^{3} - 72 x^{2} - 450 x^{3} - 60 x^{2} + 720 x + 1125 x^{2} + 150 x + 25 \cdot - 72$

Step 7.14.15

Multiply $25$ by $- 72$ .

$45 x^{4} + 6 x^{3} - 72 x^{2} - 450 x^{3} - 60 x^{2} + 720 x + 1125 x^{2} + 150 x - 1800$

Step 7.15

Subtract $450 x^{3}$ from $6 x^{3}$ .

$45 x^{4} - 444 x^{3} - 72 x^{2} - 60 x^{2} + 720 x + 1125 x^{2} + 150 x - 1800$

Step 7.16

Subtract $60 x^{2}$ from $- 72 x^{2}$ .

$45 x^{4} - 444 x^{3} - 132 x^{2} + 720 x + 1125 x^{2} + 150 x - 1800$

Step 7.17

Add $- 132 x^{2}$ and $1125 x^{2}$ .

$45 x^{4} - 444 x^{3} + 993 x^{2} + 720 x + 150 x - 1800$

Step 7.18

Add $720 x$ and $150 x$ .

$45 x^{4} - 444 x^{3} + 993 x^{2} + 870 x - 1800$