Answer
$$a = 5.36,{\text{ }}B = {40.7^ \circ },\,\,C = {78.3^ \circ }$$
Work Step by Step
$$\eqalign{
& {\text{From the triangle we have:}} \cr
& A = {61^ \circ },{\text{ }}b = 4\,{\text{ and }}\,c = 6{\text{ }} \cr
& \cr
& {\text{Use the Law of cosines to find }}a \cr
& {a^2} = {b^2} + {c^2} - 2bc\cos A \cr
& {\text{Substitute}} \cr
& {a^2} = {\left( 4 \right)^2} + {\left( 6 \right)^2} - 2\left( 4 \right)\left( 6 \right)\cos \left( {{{61}^ \circ }} \right) \cr
& {\text{Use a calculator}} \cr
& {a^2} \approx 28.72913823 \cr
& {\text{Take square roots and choose the positive root}} \cr
& a \approx 5.36 \cr
& \cr
& {\text{Calculate the angle }}B{\text{ using the law of sines}} \cr
& \frac{b}{{\sin B}} = \frac{a}{{\sin A}} \cr
& \sin B = \frac{{b\sin A}}{a} \cr
& \sin B = \frac{{4\sin \left( {{{61}^ \circ }} \right)}}{{5.36}} \cr
& {\text{Use a calculator}} \cr
& \sin B \approx 0.652701274 \cr
& {\text{Use the inverse sine function}} \cr
& B = {40.7^ \circ } \cr
& \cr
& {\text{Calculate }}C \cr
& C = {180^ \circ } - A - B \cr
& C = {180^ \circ } - {61^ \circ } - {40.7^ \circ } \cr
& C = {78.3^ \circ } \cr
& \cr
& {\text{Answer}} \cr
& a = 5.36,{\text{ }}B = {40.7^ \circ },\,\,C = {78.3^ \circ } \cr} $$
