Expand and simplify $2(5m+3n)^2$. Factorise completely $24(mn)^2 - 21mn^3$.

1. Expand the squared binomial using $(a+b)^2=a^2+2ab+b^2$.

$(5m+3n)^2=(5m)^2+2(5m)(3n)+(3n)^2$

$=25m^2+30mn+9n^2$

2. Multiply the expanded expression by 2 and simplify.

$2(5m+3n)^2=2(25m^2+30mn+9n^2)$

$=50m^2+60mn+18n^2$

1. Rewrite the first term clearly and identify the highest common factor of both terms.

$24(mn)^2-21mn^3=24m^2n^2-21mn^3$

The common factor is $3mn^2$.

2. Factor out the common factor completely.

$24m^2n^2-21mn^3=3mn^2(8m-7n)$