Let | → A _ 1 | = 3, | → A _ 2 | = 5 and | → A _ 1 + → A _ 2 | = 5...

\left|{\overset{\to}{A}}_{1} + {\overset{\to}{A}}_{2}\right| = 5$<br/>${\left|{\overset{\to}{A}}_{1} + {\overset{\to}{A}}_{2}\right|}^{2} = 25$<br/>${\left|{\overset{\to}{A}}_{1}\right|}^{2} + {\left|{\overset{\to}{A}}_{2}\right|}^{2} + 2{\overset{\to}{A}}_{1}{\overset{\to}{A}}_{2} = 25$<br/>$9 + 25 + 2{\overset{\to}{A}}_{1} ∙ {\overset{\to}{A}}_{2} = 25$<br/>${\overset{\to}{A}}_{1} . {\overset{\to}{A}}_{2} = - \dfrac{9}{2}$<br/>Now, \left(2{\overset{\to}{A}}_{1} + 3{\overset{\to}{A}}_{2}\right) ∙ \left(3{\overset{\to}{A}}_{1} - 2{\overset{\to}{A}}_{2}\right) = 6{\left|{\overset{\to}{A}}_{1}\right|}^{2} - 4{\overset{\to}{A}}_{1}{\overset{\to}{A}}_{2} + 9{\overset{\to}{A}}_{1}{\overset{\to}{A}}_{2} - 6{\left|{\overset{\to}{A}}_{2}\right|}^{2}$
$= 6 . {\left|{\overset{\to}{A}}_{1}\right|}^{2} + 5{\overset{\to}{A}}_{1}{\overset{\to}{A}}_{2} - 6{\left|{\overset{\to}{A}}_{2}\right|}^{2}$
$= 6 \times 9 + 5\left(- \dfrac{9}{2}\right) - 6 \times 25$
$= 54 - \dfrac{45}{2} - 150$
$= - 118.5$