Simplify -3a(a+b-5)+4(-2a+2b)+b(a+3b-7)

$- 3 a (a + b - 5) + 4 (- 2 a + 2 b) + b (a + 3 b - 7)$

Simplify each term.

Apply the distributive property.

$- 3 a \cdot a - 3 a b - 3 a \cdot - 5 + 4 (- 2 a + 2 b) + b (a + 3 b - 7)$

Simplify.

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$- 3 (a \cdot a) - 3 a b - 3 a \cdot - 5 + 4 (- 2 a + 2 b) + b (a + 3 b - 7)$

Multiply $a$ by $a$ .

$- 3 a^{2} - 3 a b - 3 a \cdot - 5 + 4 (- 2 a + 2 b) + b (a + 3 b - 7)$

Multiply $- 5$ by $- 3$ .

$- 3 a^{2} - 3 a b + 15 a + 4 (- 2 a + 2 b) + b (a + 3 b - 7)$

Apply the distributive property.

$- 3 a^{2} - 3 a b + 15 a + 4 (- 2 a) + 4 (2 b) + b (a + 3 b - 7)$

Multiply $- 2$ by $4$ .

$- 3 a^{2} - 3 a b + 15 a - 8 a + 4 (2 b) + b (a + 3 b - 7)$

Multiply $2$ by $4$ .

$- 3 a^{2} - 3 a b + 15 a - 8 a + 8 b + b (a + 3 b - 7)$

Apply the distributive property.

$- 3 a^{2} - 3 a b + 15 a - 8 a + 8 b + b a + b (3 b) + b \cdot - 7$

Simplify.

Rewrite using the commutative property of multiplication.

$- 3 a^{2} - 3 a b + 15 a - 8 a + 8 b + b a + 3 b \cdot b + b \cdot - 7$

Move $- 7$ to the left of $b$ .

$- 3 a^{2} - 3 a b + 15 a - 8 a + 8 b + b a + 3 b \cdot b - 7 \cdot b$

Multiply $b$ by $b$ by adding the exponents.

Move $b$ .

$- 3 a^{2} - 3 a b + 15 a - 8 a + 8 b + b a + 3 (b \cdot b) - 7 \cdot b$

Multiply $b$ by $b$ .

$- 3 a^{2} - 3 a b + 15 a - 8 a + 8 b + b a + 3 b^{2} - 7 \cdot b$

$- 3 a^{2} - 3 a b + 15 a - 8 a + 8 b + b a + 3 b^{2} - 7 b$

Add $- 3 a b$ and $b a$ .

Reorder $b$ and $a$ .

$- 3 a^{2} + 15 a - 8 a + 8 b - 3 a b + a b + 3 b^{2} - 7 b$

Add $- 3 a b$ and $a b$ .

$- 3 a^{2} + 15 a - 8 a + 8 b - 2 a b + 3 b^{2} - 7 b$

Subtract $8 a$ from $15 a$ .

$- 3 a^{2} + 7 a + 8 b - 2 a b + 3 b^{2} - 7 b$

Subtract $7 b$ from $8 b$ .

$- 3 a^{2} + 7 a + b - 2 a b + 3 b^{2}$

Reorder.

Move $b$ .

$- 3 a^{2} + 7 a - 2 a b + 3 b^{2} + b$

Move $7 a$ .

$- 3 a^{2} - 2 a b + 3 b^{2} + 7 a + b$