Simplify (m-2n)(3m+5n)+4mn

$(m - 2 n) (3 m + 5 n) + 4 m n$

Simplify each term.

Expand $(m - 2 n) (3 m + 5 n)$ using the FOIL Method.

Apply the distributive property.

$m (3 m + 5 n) - 2 n (3 m + 5 n) + 4 m n$

Apply the distributive property.

$m (3 m) + m (5 n) - 2 n (3 m + 5 n) + 4 m n$

Apply the distributive property.

$m (3 m) + m (5 n) - 2 n (3 m) - 2 n (5 n) + 4 m n$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 m \cdot m + m (5 n) - 2 n (3 m) - 2 n (5 n) + 4 m n$

Multiply $m$ by $m$ by adding the exponents.

Move $m$ .

$3 (m \cdot m) + m (5 n) - 2 n (3 m) - 2 n (5 n) + 4 m n$

Multiply $m$ by $m$ .

$3 m^{2} + m (5 n) - 2 n (3 m) - 2 n (5 n) + 4 m n$

Rewrite using the commutative property of multiplication.

$3 m^{2} + 5 m n - 2 n (3 m) - 2 n (5 n) + 4 m n$

Rewrite using the commutative property of multiplication.

$3 m^{2} + 5 m n - 2 \cdot 3 n m - 2 n (5 n) + 4 m n$

Multiply $- 2$ by $3$ .

$3 m^{2} + 5 m n - 6 n m - 2 n (5 n) + 4 m n$

Rewrite using the commutative property of multiplication.

$3 m^{2} + 5 m n - 6 n m - 2 \cdot 5 n \cdot n + 4 m n$

Multiply $n$ by $n$ by adding the exponents.

Move $n$ .

$3 m^{2} + 5 m n - 6 n m - 2 \cdot 5 (n \cdot n) + 4 m n$

Multiply $n$ by $n$ .

$3 m^{2} + 5 m n - 6 n m - 2 \cdot 5 n^{2} + 4 m n$

Multiply $- 2$ by $5$ .

$3 m^{2} + 5 m n - 6 n m - 10 n^{2} + 4 m n$

Subtract $6 n m$ from $5 m n$ .

Move $n$ .

$3 m^{2} + 5 m n - 6 m n - 10 n^{2} + 4 m n$

Subtract $6 m n$ from $5 m n$ .

$3 m^{2} - m n - 10 n^{2} + 4 m n$

Add $- m n$ and $4 m n$ .

$3 m^{2} - 10 n^{2} + 3 m n$